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Instituto de Matemáticas Aplicadas UCV

Caleta Numérica

Caleta Numérica

Seminar on Numerical Analysis and Applied Mathematics at Valparaíso

Seminar on Numerical Analysis and Applied Mathematics at Valparaíso.

The seminar takes place on fridays at 16:00 in Room 2-1, Instituto de Matemáticas, PUCV.

CONTACT

Diego Paredes

email: diego.paredes@ucv.cl
phone: (+56 32) 2274019
webpage: http://ima.ucv.cl/seminarios/caleta-numerica/


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Instituto de Matemáticas

Pontificia Universidad Católica de Valparaíso

Blanco Viel 596

Cerro Barón

Valparaíso

Chile

Calendario 2016

Noviembre

25 - Norbert Heuer (PUC Chile)

18 - Erwin Hernández (Universidad Santa María, Chile)

11 - Dae-Jin Lee (BCAM, Spain)

Septiembre

30 - Diego Paredes (IMA - PUCV)

23 - Jorge Clarke (Universidad Santa María)

09 - Gino Montecinos (CMM, Universidad de Chile)

Agosto

19 - Elisabete Alberdi (Universidad del País Vasco)

Mayo

13 - Michael Karkulik (Universidad Santa María)

06 - Vincent Darrigrand (Universidad del Pais Vasco & INRIA-Magique 3D, Université de Pau)

Marzo

11 - Gabriel Barrenecha (University of Strathclyde)

Calendario 2015

Agosto

07 - Nicole Spillane (Centro de Modelación Matemática - Universidad de Chile)

Julio

10 - Kris Van der Zee (The University of Nottingham)

03 - Antônio Tadeu Azevedo Gomes (Laboratório Nacional de Computação Científica - BRASIL)

Junio

12 - Thomas Führer (Pontificia Universidad Católica de Chile)

05 - Michael Karkulik (Pontificia Universidad Católica de Chile)

Mayo

29 - Jessika Camaño (Universidad Católica de la Santísima Concepción)

08 - Maciej Paszynski (AGH University of Science and Technology)

Abril

17 - Gabriel Barrenechea (University of Strathclyde)

10 - Claudio Torres (Universidad Técnica Federico Santa María)

Marzo

13 - Ivan Graham (University of Bath)

Calendario 2014

Diciembre

12 - Felipe Lepe (Universidad de Concepción)

Noviembre

21 - Manuel Solano (Universidad de Concepción)

14 - Raúl Fierro (IMA - PUCV)

07 - Karin Saavedra (Universidad de Talca)

Octubre

24 - Kris Van der Zee (The University of Nottingham)

17 - Tomás Barrios (Universidad Católica de la Santísima Concepción)

10 - Ricardo Oyarzúa (Universidad del Bío-Bío)

Septiembre

26 - Richard Rankin (Universidad Técnica Federico Santa María)

Agosto

22 - Diego Paredes (IMA-PUCV)

Junio

20 - Jay Gopalakrishnan (Portland State University, USA)

Mayo

02 - Karina Vilches (Universidad de Chile)

Abril

04 - Felipe Millar (Universidad del Bío-Bío)

Marzo

28 - Paulina Sepúlveda Salas (Portland State University, USA)

Calendario 2013

Noviembre

15 - Luis Briceño (Universidad Técnica Federico Santa María)

Octubre

25 - Joaquín Mura (Ingeniería Civil - PUCV)

11 - Sebastián Ossandón (IMA - PUCV)

Septiembre

27 - Loredana Smaranda (Universitatea din Pitesti, Rumania)

13 - David Pardo (Universidad del País Vasco & Ikerbasque, España)

Upcoming Seminars

Noviembre 25, 15:40, Norbert Heuer, Pontificia Universidad Católica de Chile

Recent advances in the DPG method

We will present recent advances for a variant of the discontinuous Petrov-Galerkin (DPG) method which uses so-called optimal test functions.

The aim of the method is to guarantee robust stability of approximations for different kinds of problems based on partial differential equations.

We will start with a simple introduction to illustrate what stability means in the case of systems of linear equations, and how it is generalized to variational formulations. We then explain what this has to do with DPG analysis and how the analysis of adjoint problems enters.

As an illustration we consider the heat equation for which we present a combination of DPG with time stepping. Specifically, we use the backward Euler scheme as time discretisation and propose a DPG space approximation of the time-discrete scheme. Well-posedness and stable approximation properties are obtained from a precise analysis of the underlying time-discrete variational formulation at every time step. Appropriate convergence properties for field variables are deduced. We present numerical experiments that underline our theoretical results.

This is joint work with Thomas Fuehrer and Jhuma Sen Gupta and has been supported by CONICYT-Chile through Fondecyt grants 1150056, 3150012, and Anillo ACT1118 (ANANUM).

Past Seminars

Noviembre 18, Erwin Hernández, Universidad Técnica Federico Santa María

Análisis Numérico y computacional de problemas de control de vigas y vibraciones elastoacústicas

En esta charla revisaremos el control de propiedades mecánicas de estructuras, así como su comportamiento en sistemas acoplados. Consideramos tanto estructuras delgadas como solas o inmersas en un fluido.

Para comprender el fenómeno en el caso de una estructura delgada, presentaremos un esquema libre de bloqueo para resolver las ecuaciones de vigas y resolveremos el problema de localización de fuerzas que actúan sobre la viga para controlar su deflexión cuando es sometida a fuerzas externas. Veremos también, una técnica de control pasivo para un problema exterior de vibraciones elastoacústicas. Mostraremos resultados numéricos de los esquemas considerados.

 

 

 

 

 

 

 

 

 

Noviembre 11, Dae-Jin Lee, Basque Center for Applied Mathematics (BCAM)

Splines con penalizaciones: metodología y aplicaciones

Los splines con penalizaciones (o P-splines) han alcanzado una gran popularidad en los últimos 20 años en diversos problemas de regresión y suavizado. Su principal ventaja es la flexibilidad a la hora de modelizar datos de diferente naturaleza y su ganancia computational con respecto a otros métodos. En esta charla se presentará una revisión de la metodología desarrollada, extensiones y aplicaciones a datos reales.

Septiembre 30, Diego Paredes, IMA-PUCV

Métodos de elementos finitos mixto-híbridos multiescala

 

La nueva familia de métodos de elementos finitos mixto-híbridos multiescala (MHM), tiene como objetivo resolver problemas de contorno con comportamiento multiescala usando mallas gruesas. El proceso de upscaling utilizado transfiere para las funciones de base la responsabilidad de alcanzar una precisión de mayor orden. Esto se logra us- ando un marco general de hibridización en el que se relaja la continuidad de la solución y esta es impuesta débilmente a través de la acción de multiplicadores de Lagrange. En esta charla la metodología MHM es aplicada a diferentes problemas. De esta forma se caracteriza las incógnitas como soluciones de problemas locales con condiciones de contorno dirigidas por los multiplicadores de Lagrange. Tales problemas locales son independientes entre sí lo que induce naturalmente una implementación paralela.

 

 

 

 

Septiembre 23, Jorge Clarke, Universidad Santa María

Regularity properties and simulation of Gaussian random fields on

 

In many geosciences applications, the phenomena of interest are observed on large portion of the Planet Earth. When such phenomena evolves over time, a valid model for the study of these observations is to consider them as partial realization of a spatio- temporal random field, where the spatial component is defined on a sphere, considering this last as a more realistic representation of the globe. We Introduce a family of Gaussian random fields constructed by spectral methods via a double Karhunen-Loéve type representation. Based on later result on the literature we claim that this family represent the class of Isotropic and stationary Gaussian random fields over S^d × R. In the formulation, we exploit the theory about positive definite functions on the sphere and its extension to the spatio-temporal context. We then study regularity properties of the proposed family via Sobolev spaces and interpolation techniques. Then we approximate the elements of this family with a double truncation and we analyse the accuracy of this technique evaluating the error in L2-norm and through simulation experiments.

 

 

 

 

Septiembre 9, Gino Montecinos, CMM, Universidad de Chile

A brief review of ADER, a family of high order finite volume schemes

The Godunov theorem provides a limitation for the accuracy of finite volume schemes, linear and monotone; only first order of accuracy can be achieved. A family of non-linear schemes, called ADER (Arbitrary accuracy DERivative Riemann problems), which circumvents Godunov’s theorem, was proposed by E.F. Toro and collaborators (2001-2002). Building block for these schemes are: the polynomial reconstruction of cell averages and; the solution of special Cauchy problem, the so called generalized Riemann problems. The reconstruction and the solution of Generalized Riemann problems allow us to construct schemes of accuracy p > 1 such that spurious oscillations near high gradients are controlled. Advantages of these finite volume schemes are: efficiency, stability and accuracy. In this seminar are presented: some notions on hyperbolic problems and elementary waves; classical Riemann problems and Godunov’s type schemes; the Godunov theorem and the Harten theorem, which are the main results regarding the construction of numerical schemes of ADER type; the generalized Riemann problems and the construction of high-order finite volume methods.

 

 

Agosto 19, Elisabete Alberdi, Universidad del Pais Vasco

Designing techniques to find Symmetric Composition Methods of Symmetric Integrators

Composition methods are useful when solving Ordinary Differential Equations (ODEs) as they increase the order of accuracy of a given basic numerical integration scheme. We will focus on symmetric composition methods involving some basic second order symmetric integrator with different step sizes [1]. The introduction of symmetries into these methods simplifies the order conditions and reduces the number of unknowns. Several authors have worked in the search of the coefficients of these type of methods: the best method of order 8 has 17 stages [2], methods of order 8 and 15 stages were given in [3, 5, 6], 10-order methods of 31, 33 and 35 stages have been also found [2, 4]. In this work a technique that we have built to obtain 10-order symmetric composition methods of symmetric integrators of s = 31 stages (16 order conditions) is explained. Given some starting coefficients that satisfy the simplest five order conditions, the process followed to obtain the coefficients that satisfy the sixteen order conditions is provided.

 

 

 

 

 

Mayo 13, Michael Karkulik, Universidad Santa María

Space-time least squares methods for evolution equations

We consider two types of evolution equations: (i) linear second-order parabolic problems (e.g., the heat equation), and (ii), linear first-order hyperbolic systems (e.g., the wave equation). Classical numerical schemes for such equations discretize first in space and use ODE-techniques to advance in time. In our talk, we present variational formulations in space-time. This means that we do not treat the time as an extra dimension, but discretize and solve in the whole space-time cylinder. Our variational formulations are of least-squares type. We show existence and uniqueness of conforming finite element solutions and a-priori error estimates. This is joint work with Jay Gopalakrishnan, Martin Neumueller, and Panayot Vassilevski.

 

 

Mayo 6, Vincent Darrigrand, Universidad del Pais Vasco & INIRIA-Magique 3D Université de Pau

Generalised Error Representations for Goal­-Oriented Adaptivity

In the scope of subsurface modeling via the resolution of inverse problems, the so­called goal­ oriented adaptivity plays a fundamental role. Indeed, while classical adaptive algorithms were first designed to accurately approximate the energy norm of a problem, in geophysical problems one often requires a good approximation of a specific quantity of interest. The goal­oriented approach consists in expressing the error in the quantity of interest as an integral over the entire computational domain involving the errors of the original and adjoint problems, and then minimise an upper bound of such error representation by performing local refinements.

Most authors use the adjoint problem to represent the approximation error of a quantity of interest via the global bilinear form that describes the problem in terms of local and computable quantities. Our methodology, however, is based on the selection of an alternative bilinear form exhibiting better properties than the original bilinear form (e.g. positive definiteness). We represent the residual error functional of the adjoint problem through this alternative form using the Riesz representation. We can then compute new upper bounds of the error of a quantity of interest in a similar way than with the classical approach.

Our main contribution is to demonstrate that a proper choice of such alternative form may improve the upper bounds of the error representation. Moreover, the method proposed here generalises the existing ones, since in particular, we can select as the alternative bilinear form the one associated to the adjoint problem. The proposed method can be applied to rather general 1D, 2D and 3D problems. Our upper bounds are sharper than the classical ones if the alternative operator is wisely selected.

We describe numerically the main features and limitations of the proposed method with 1D, 2D and 3D Helmholtz examples. Extensive numerical results are illustrated using uniform h­ and p­ refinements, as well as a simple self­adaptive goal­oriented p­refinement strategy. The application to other adaptive algorithms such as a goal­oriented hp­adaptive algorithm, adaptivity in a high continuity space, or adaptivity in time domain is straightforward.

 

 

Marzo 11, Gabriel Barrenecha, University of Strathclyde

A unified analysis of algebraic flux-correction schemes

In this talk I will review recent results on the mathematical analysis of algebraic flux-correction (AFC) schemes. The schemes are designed mainly to preserve a discrete version of the maximum principle, and, unlike usual staibilised finite element schemes, AFC schemes do not start from a weak formulation, but rather they only “see” the linear system resulting from the discretisation. This lack of weak formulation is one of the main issues that have prevented from providing a mathematical analysis for this class of schemes. Then, the first step is to to write this scheme in a weak form, and then obtain stability results. Positivity of the scheme is proved under some approriate conditions on the mesh, and convergence results are proved.

 

Agosto 7, Nicole Spillane, Centro de Modelamiento Matemático

Achieving Robustness in Domain Decomposition

Domain Decomposition methods are a family of solvers that aim to solver very large linear systems on parallel architectures. They proceed by splitting the computational domain into subdomains and then approximating the inverse of the original problem by a sum of local inverses in the subdomains. I will present a classical domain decomposition method and show that for realistic simulations (with heterogeneous materials for instance) convergence can become very slow. Then I will explain how this can be fixed by using either a coarse space (this is also known as deflation) or multiple search directions within the conjugate gradient algorithm.

 

 

 

 

 

 

Julio 10, K.G. van de Zee, The University of Nottingham

Fundamentals of goal-oriented error estimation and adaptivity

Goal-oriented computational methodologies are extremely useful in the numerical solution of partial differential equations (PDEs) in the engineering and applied sciences. The idea of a goal oriented computation embraces the notion that a numerical simulation is generally done to study specific features of the solution, instead of the solution itself. These features are the so-called quantities of interest or target outputs, which are, therefore, the true goals of the simulation.

In this talk I will focus on the adaptive approximation of PDEs using goal-oriented (or adjoint-based, or duality-based) error estimates. I will start by recalling fundamental theory of goal-oriented a posteriori error estimation, and subsequently present a recent abstract convergence analysis of an optimal goal oriented adaptive finite element method.

 

 

 

 

 

 

Julio 3, Antônio Gomes (Laboratório Nacional de Computação Científica – BRASIL)

Experiments with Multi-Language Approaches to Implementing Scalable FEM/FVM solvers

 

In this talk we discuss about approaches to the development of scalable FEM/FVM solvers based on multiple programming languages. The aim of this type of “multi-language approach” is to cater for different software quality attributes (performance, fault tolerance, configurability, maintainability) by adequately exploring the features of different programming languages. Such an approach has been used elsewhere (e.g. the FENICS project) and has been also explored in a joint project involving PUCV, LNCC (Brazil) and INRIA (France). In this project we have progressed work on the implementation of a solver specifically crafted for a new family of FE methods called Multiscale Hybrid-Mixed (MHM). The MHM method allows solving (global) problems on coarse meshes while providing solutions with high-order precision by exploring the loosely-coupled strategy of embedding independent (local) subproblems in the upscaling procedure. Sticking to the idea of multi-language approaches, we’ve adopted Erlang as the base implementation for the communicating processes of the new solver. In this talk we present the advantages of adopting Erlang for such processes in terms of high productivity, good scalability, and support to fault tolerance. We show how the Erlang processes are loosely integrated with numerical computing processes (implemented in C++) that ultimately solve the global and local problems, thus indicating the potential of the Erlang implementation in being adopted in other contexts, e.g. for scalable FVM solvers. We present a preliminary performance evaluation of the overall (Erlang plus C++) implementation, taking into account its speed-up and load balancing properties. We then conclude the talk by pointing out the intended future work.

 

 

Junio 12, Thomas Führer (Pontificia Universidad Católica)

A wirebasket preconditioner for the mortar boundary element method

 

We present and analyze a preconditioner of the additive Schwarz type for the mortar boundary element method. As a basic splitting, on each subdomain we separate the degrees of freedom related to its boundary from the inner degrees of freedom. The corresponding wirebasket type space decomposition is stable up to logarithmic terms. For the blocks that correspond to the inner degrees of freedom standard preconditioners for the hypersingular integral operator on open boundaries can be used. For the boundary and interface parts as well as the Lagrangian multiplier space, simple diagonal preconditioners are optimal. Our technique applies to quasi-uniform and non-uniform meshes of shape-regular elements. Numerical experiments on triangular and quadrilateral meshes confirm theoretical bounds for condition and MINRES iteration numbers.

 

 

Junio 5, Michael Karkulik (Pontificia Universidad Católica)

DPG method with optimal test functions for a transmission problem

We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultraweak finite element formulation on the interior. To be precise, it is a nonsymmetric coupling as developed in [3], but with an ultraweak finite element part as used in [1]. To guarantee the discrete inf-sup condition, we discretize the whole system with optimal test functions. In this way, we obtain quasi-optimality of the method in the so-called energy norm. In order to relate the energy norm to usual Sobolev norms, we use recent results regarding the nonsymmetric coupling of finite elements and boundary elements on polygonal interfaces, cf. [2, 4].

[1] L. Demkowicz and J. Gopalakrishnan. Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal., 49(5):1788–1809, 2011.

[2] G. N. Gatica, G. C. Hsiao, and F.-J. Sayas. Relaxing the hypotheses of Bielak-MacCamy’s BEM-FEM coupling. Numer. Math., 120(3):465–487, 2012.

[3] C. Johnson and J.-C. Nédélec. On the coupling of boundary integral and finite element methods. Math. Comp., 35(152):1063–1079, 1980.

[4] F.-J. Sayas. The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces. SIAM Rev., 55(1):131–146, 2013

 

 

 

Mayo 29, Jessika Camaño (Universidad Católica de la Santísima Concepción)

A new augmented mixed finite element method for the Navier-Stokes problem

Abstract

In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinear-pseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear pseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach’s fixed point Theorem and Lax-Milgram’s Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well posedness and the corresponding Cea’s estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear pseudostress tensor, and continuous piecewise polynomial elements of degree k+1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.

[1] Z. Cai and S. Zhang, Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation, Mathematics of Computations, vol. 81, no. 280, 1903-1927, (2012).

[2] J. Camaño, R. Oyarzúa and G. Tierra, Analysis of an augmented mixed-FEM for the Navier Stokes problem, Preprint CI2MA 2014-33, Universidad de Concepción, Concepción, 2014.

[3] G.N. Gatica, A. Márquez and M.A. Sánchez, Analysis of a velocity-pressure- pseudostress formulation for incompressible flow, Computer Methods in Applied Me- chanics and Engineering, vol. 199, 17-20, pp. 1064-1079, (2010).

[4] J. S. Howell and N. Walkington, Dual Mixed Finite Element Methods for the Navier Stokes Equations. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 47, 789-805, (2013).

 

Partially supported by CONICYT-Chile through project Inserción de Humano Avanzado en la Academia 79130048; pro ject Fondecyt 11140691

 

 

 

 

Mayo 4, Maciej Paszynski (AGH University of Science and Technology, Krakow, Poland)

Parallel alternating direction preconditioner for isogeometric simulations of explicit dynamics

Abstract

In this talk a parallel implementation of the alternating direction preconditioner for isogeometric simulations of explicit dynamics is presented. The Alternating Direction Implicit (ADI) algorithm, belongs to the category of matrix-splitting iterative methods [1–4]. The new version of this algorithm has been recently developed for isogeometric simulations of two dimensional explicit dynamics [5] and steady-state diffusion equations with orthotropic heterogenous coefficients [6]. In this talk we present a parallel version of the alternating direction implicit algorithm for three dimensional simulations. The algorithm has been incorporated as a part of PETIGA an isogeometric framework [7] build on top of PETSc [8]. We show the scalability of the parallel algorithm on STAMPEDE linux cluster up to 10,000 processors, as well as the convergence rate of the PCG solver with ADI algorithm as preconditioner. We conclude the presentation with simulation of the non linear flow in heterogenous media [9], as well as simulation deformation of solid object by extrnal forces [10]. We show that the solver presents an efficient alternative for classical direct solvers [11].

[1] D.W. Peaceman, H.H. Rachford Jr., The numerical solution of parabolic and elliptic
differential equations, Journal of the Society for Industrial and Applied Mathematics, 3 (1955)
28–41.
[2] J. Douglas, H. Rachford, On the numerical solution of heat conduction problems in two and
three space variables, Transactions of the American Mathematical Society, 82 (1956)
421–439.
[3] E.L. Wachspress, G. Habetler, An alternating-direction-implicit iteration technique, Journal of
the Society for Industrial and Applied Mathematics, 8 (1960) 403–423.
[4] G. Birkhoff, R.S. Varga, D. Young, Alternating direction implicit methods, Advanced
Computing 3 (1962) 189–273.[5] L. Gao, V. M. Calo, Fast Isogeometric Solvers for Explicit
Dynamics, Computer Methods in Applied Mechanics and Engineering, 274 (2014) 19-41.
[6] L. Gao, V. M. Calo, Preconditioners based on the alternating-direction-implicit algorithm for
the 2D steady-state diffusion equation with orthotropic heterogeneous coefficients, Journal of
Computational and Applied Mathematics, 01 (2015) 274-295.
[7] N. Collier, L. Dalcin, V. Calo, PetIGA: High- Performance Isogeometric Analysis,
arXiv:1305.4452 [cs.MS]
[8] S. Balay. S. Abhyankar. M. F. Adams, J. Brown, P. Brune, K. Buschelman, V Eijkhout, W. D.
Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, H. Zhang, PETSc User
Manual, http://www.mcs.anl.gov/petsc (2014)
[9] Maciej Wozniak, Marcin Los, Maciej Paszynski, Lisandro Dalcin, Victor Calo, Parallel fast isogeometric solvers for explicit dynamics accepted to Computing and Informationcs (2015)
[10] M. Los, M. Wozniak, M. Paszynski, L. Dalcin, and V.M. Calo, Dynamics with Matrices Possessing Kronecker Product
Structure, accepted to Procedia Computer Science (2015)
[11] Maciej Wozniak, Maciej Paszynski, David Pardo, Lisandro Dalcin, Victor Calo, Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines Computer Methods in Applied Mechanics and Engineering, 284 (2015) 971-987.

Collaborators: Marcin Los, Maciej Wozniak, Lisandro Dalcin, Victor Calo

 

 

 

 

Abril 17, Gabriel Barrenechea (University of Strathclyde)

Stabilising some inf-sup stable pairs on anisotropic quadrilateral meshes

Abstract

The finite element solution of the Stokes problem is subject to the well-known inf-sup condition. This condition usually depends on the domain and the degree of the polynomial spaces used in the discretisation. Now, when anisotropic finite elements are used, this inf-sup condition usually degenerates with the aspect ratio. In this talk I will present results identifying the part of the pressure space that is responsable for this degeneration, and present a way to solve that. In the second part of the talk, this technique will be applied to some, non inf-sup stable this time, pairs of spaces for the Stokes and Oseen equations. The work presented in this talk is in collaboration with Mark Ainsworth (Brown) and Andreas Wachtel (Strathclyde).

 

 

 

 

Abril 10, Claudio Torres (Universidad Técnica Federico Santa María)

An extension of a simple vertex model for mathematical modeling of grain growth in polycrystalline materials

Abstract

The present talk will discuss past and current efforts associated with the modeling of polycrystalline material undergoing coarsening in 2D. The polycrystalline material is modeled as a graph where the vertices and edges describe the grain network. The challenge comes from the description of the dynamics of the process. Traditionally, two main approaches have been used to describe the evolution o the grain boundaries (edges) and triple-junctions (vertices). The first approach consider that the dominant part of the dynamic is controlled by the edges (curvature driven model) and the second approach consider that the dominant part of the evolution is controlled by the vertices (vertex model). We will discussed in detail the, so called, vertex model to give mathematical foundation to the topological transitions needed for the evolution, these are: flipping and grain removal. We will finish with a possible extension of the model and its coupling with curvature driven models.

 

 

 

 

Marzo 13, Ivan Graham (University of Bath)

On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption

Abstract

As a model problem for high-frequency wave scattering, we study both the interior impedance problem for the Helmholtz equation and the truncated sound soft scattering problem. Finite element approximations of this problem for high wavenumber are notoriously hard to solve. The analysis of solvers such as GMRES is also hard, since the corresponding system matrices are complex, non-Hermitian and usually highly non-normal and so information about spectra and condition numbers generally does not give much information about the convergence rate.

We consider preconditioners built from approximating the corresponding PDE problem with added absorption (the “shifted Laplace problem”). Using techniques from PDE analysis we show how the absorption should be chosen so as to obtain wavenmuber independent GMRES convergence for the preconditioned problem. We also present an analysis of additive Schwarz domain decomposition methods, proving estimates on the rate of convergence explicitly in terms of wavenumber and absorption. We give numerical illustrations of the performance of the solver for some constant and variable wavespeed problems.

The optimal choice of absorption is studied experimentally.

The talk includes joint work with Martin Gander (Geneva), Euan Spence (Bath) and Eero Vainikko (Tartu).

 

 

 

 

Diciembre 12, Felipe Lepe (Universidad de Concepción)

Finite Element Analysis for a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

Abstract

In this work we analyze a mixed finite element method for approximating the vibration frequencies and modes of a non-homogeneous Timoshenko beam. Optimal order error estimates are proved for displacements, rotations, shear stress and the bending moments of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are independent of the beam thickness, which lead to the conclusion that the method is locking free. Finally, we report numerical experiments which allow us to assess the performance of the method.

 

 

Noviembre 21, Manuel Solano (Universidad de Concepción)

Hybridizable discontinuous Galerkin method on general domains

Abstract

We present a technique for numerically solving Dirichlet boundary-value problems in domains with curved boundary. The technique consists in approximating the domain by polyhedral subdomains where a hybridizable discontinuous Galerkin (HDG) method is used to solve for the approximate solution. The approximation is then suitably extended to the remaining part of the domain. This approach allows for the use of only polyhedral elements and there is no need of fitting the boundary in order to obtain an accurate approximation of the solution. Optimal error estimates are obtained. Moreover we numerically explore the robustness of the method with respect to distance between the boundary and the computational domain.

 

 

 

Noviembre 14, Raúl Fierro (IMA-PUCV)

Approximation of Some Discrete-Time Stochastic Processes by Differential Equations

Abstract

This work deals with solutions of ordinary differential equations as approximations of some discrete-time stochastic processes. Similarly, these stochastic processes may be seen as schemes of approximation for this solution. Indeed, these stochastic schemes are defined and their convergence to the solution of a differential equation is proven. Moreover, the asymptotic distribution of the fluctuations about the limit solution is studied. This fact gives the asymptotic distribution of a random global error of approximation. Main results are illustrated by means of the so called SIS epidemic model and numerical simulations are carried out.

 

 

 

Noviembre 7, Karin Saavedra (Universidad de Talca)

Nuevas contribuciones a una estrategia multiescala de descomposición de dominios para la simulación de materiales compuestos laminados

Resumen

La utilización de simulaciones numéricas como herramienta para disminuir los costos asociados a la utilización de materiales compuestos sigue siendo un desafío en la actualidad, principalmente, debido a la existencia de fenómenos no lineales y a la necesidad de discretizaciones finas de elementos finitos. Para contribuir a esta problemática, el autor ha propuesto en trabajos anteriores una estrategia de descomposición de dominios multiescala para modelar la interacción entre pandeo, delaminación y contacto en materiales compuestos laminados. Estas simulaciones cuentan con dos escalas de resolución: la microescala que está asociada a la discretización fina de cada sub-estructura y la macroescala que asegura la propagación de la parte de la solución con gran variación de onda. En esta exposición se presentarán los principios de la estrategia y algunos progresos que se han realizado recientemente en la macroescala para asegurar una rápida convergencia del proceso iterativo. (Trabajo financiado por CONICYT: Fondecyt de Iniciación n° 11130623)

 

 

 

 

Octubre 24, Kris van der Zee (The University of Nottingham)

Diffuse-interface tumour growth modelling, analysis and simulation

Abstract

In this talk I will consider a class of nonlinear PDEs relevant to the growth of cancerous tumours, described with so-called diffuse interfaces.

Its thermomechanical foundations will be discussed, as well as its sharp-interface limit.

I will furthermore illuminate the underlying gradient-flow structure,

which is important for the design of stable time-stepping schemes.

Numerical examples will be presented for a recently proposed 2nd-order stabilised convex-splitting scheme.

 

 

 

 

Octubre 17, Tomás Barrios (Universidad Católica de la Santísima Concepción)

Esquemas numericos de orden bajo para las ecuaciones de Reissner-Mindlin

Resumen

En esta charla se presentarán nuevos esquemas estabilizados que permiten aproximar las ecuaciones de Reissner-Mindlin. A modo de motivación, previa a la presentacion de los esquemas, utilizamos las ecuaciones de la elasticidad lineal para materiales isotrópicos, a fin de describir algunos clásicos fenómenos de bloqueo numérico, además de explicar una forma de evitarlos. Esta descripción nos sirvió de motivación para iniciar nuestro estudio a las ecuaciones de Reissner-Mindlin, con lo cual pensamos que es el medio con el cual mejor se presentan los nuevos resultados.

 

 

 

 

Octubre 10, Ricardo Oyarzúa (Universidad del Bio Bio)

A priori and a posteriori error analysis for a vorticity-based mixed formulation of the Brinkman problem

Abstact

This work deals with the analysis of new mixed finite element methods for the generalized Stokes problem formulated in terms of velocity, vorticity and pressure, with non-standard boundary conditions. By employing an extension of the Babuška-Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, on the one hand we show that Raviart-Thomas elements of order k >= 0 for the approximation of the velocity field, piecewise continuous polynomials of degree k+1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi-Douglas-Marini elements of order k+1 for the approximation of velocity, piecewise continuous polynomials of degree k+2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms and we carry out the reliability and efficiency analysis of a residual-based a posteriori error estimator. Finally, we report several numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical results on unstructured meshes. Additional examples of cases not covered by our theory are also presented.

 

 

 

Septiembre 26, Richard Rankin (Universidad Técnica Federico Santa María)

Computable Error Bounds for Finite Element Approximations of Linear Elasticity Problems

Abstract

Finite element approximation schemes can be used to obtain approximate solutions to partial differential equations. When using a numerical method to approximate the solution to any problem it is important to be able to say how accurate the approximation obtained is. We will discuss the a posteriori error estimators that we have obtained for bounding the energy norm of the error in finite element approximations of linear elasticity problems in two and three dimensions.

 

 

 

Agosto 22, Diego Paredes (IMA-PUCV)

El Método de Elementos Finitos Mixto Híbrido Multiescala.

Abstract

La nueva familia de métodos de elementos finitos, llamados métodos mixtos híbridos multiescala (o MHM), tiene como objetivo resolver problemas de contorno con comportamiento multiescala usando mallas gruesas. El proceso de upscaling utilizado, transfiere hacia las funciones de base la responsabilidad de alcanzar una precisión de orden superior. Esto se logra usando un marco general de hibridización, en el que se relaja la continuidad de la solución y ésta es impuesta débilmente a través de la acción de multiplicadores de Lagrange. De esta forma se caracterizan las incógnitas como soluciones de problemas locales con condiciones de contorno dirigidas por los multiplicadores de Lagrange. Tales problemas locales son independientes entre sí, permitiendo un proceso de paralelización natural. Presentamos validaciones numéricas para respaldar los resultados teóricos expuestos.

 

Junio 20, Jay Gopalakrishnan (Portland State University)

A Posteriori error control for DPG methods.

Abstract

We begin with a quick introduction of the DPG (discontinuous Petrov Galerkin) method, after which we present an a posteriori error analysis of the method under an abstract set of assumptions. It is possible to prove that a locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors. Since the error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete spaces of the functional framework. We then apply these ideas to a new DPG method for Stokes flow which yields an approximation to the true symmetric fluid stress in addition to other variables of interest.

 

 

Mayo 02, Karina Vilches Ponce (DIM-Universidad de Chile)

Preguntas abiertas en un sistema parabólico-elíptico del tipo Patlak-Keller-Segel: una aproximación numérica.

Abstact

Es esta ocasión presentaré algunas de las preguntas abiertas que se formularon a partir de mi trabajo de tesis doctoral, en el cual fue estudiada la condición óptima de existencia global y blowup sobre las masas iniciales para un sistema parabólico-elíptico del tipo Patlak-Keller-Segel. Veremos en detalle la formulación de la problemática y analizaremos el interés que tiene completar la búsqueda de la condición sharp, indicando claramente en qué pueden ser útiles los métodos numéricos. Finalmente, podremos conjeturar algunos posibles caminos a tomar que puedan ayudar a dar una respuesta a la problemática mostrada.

 

 

 

 

Abril 04, Felipe Millar (Universidad del Bío-Bío, Chile)

Un método de elementos finitos para el problema de pandeo de placas Kirchhoff simplemente apoyada

Abstact

El método de elementos finitos para la aproximación de problemas de valores propios es un objeto de gran interés tanto desde el punto de vista teórico como práctico. El presente trabajo tiene relación con la estabilidad elástica de placas, en particular el llamado problema de pandeo, el cual frecuentemente aparece en problemas de ingeniería asociados a puentes, naves y diseño aeroespacial. Dicho problema puede ser formulado como un problema espectral cuya solución está relacionada con el límite de la estabilidad elástica de la placa. El método propuesto está basado en una formulación introducida por Ciarlet y Raviart, y consiste en la introducción de una nueva variable σ = ∆u (donde u representa el desplazamiento transversal de la superficie media). Para el análisis de este problema a nivel continuo, se introduce un operador llamado solución que tiene directa relación con los valores propios asociados al problema de pandeo. Luego, a través del uso de la teoría de Babuska-Osborn, intentaremos caracterizar la convergencia de los valores propios y las autofunciones asociadas, mostrando una convergencia optimal.

 

 

 

 

Marzo 28, Paulina Sepúlveda Salas (Fariborz Maseeh Department of Mathematics + Statistics, Portland State University, Oregon, USA.)

Solución de problemas de propagación de ondas sísmicas utilizando CLAWPACK

Abstact

Este trabajo se basa en el modelamiento de fenómenos de propagación de ondas sísmicas a través de diferentes materiales. Usando las ecuaciones de la elastodinámica y el métodos de volúmenes finitos, se obtiene una solución numérica al problema de propagación de ondas elásticas considerando discontinuidades en los parámetros de Lamé.

La implementación de algoritmos para encontrar estas soluciones pueden ser halladas en el paquete de sofware Clawpack. En esta presentación, tomando como base la teoría, incluiré el funcionamiento y utilización de Clawpack como una herramienta para obtenerlas.

Muchos de los fenómenos de interés práctico involucran geometrías irregulares, por lo que a partir del procedimiento aplicado en mallados uniformes, se pueden obtener resultados a problemas en mallados de tipo no uniformes.

 

November 22, Luis Briceño (Departamento de Matemáticas – UTFSM, Chile)

A Monotone[latex]+[/latex]Skew Splitting Model for Composite Monotone Inclusions in Duality

Abstract

 

 

 

October 25, Joaquín Mura (Ingeniería Civil -PUCV, Chile)

Estimación de la Porosidad en un Medio Incompresible con Microburbujas a través de Mediciones No-Invasivas.

En el presente trabajo, nos interesamos en materiales compuestos por un medio elástico incompresible e inclusiones gaseosas confinadas en pequeñas burbujas. Este modelo puede representar la dinámica de tejidos blandos, como por ejemplo el pulmonar. Para visualizar el campo de porosidad en el medio nos apoyamos en técnicas de elastografía por resonancia magnética, de donde se puede obtener información acerca del estado de deformaciones del espécimen en estudio. Considerando un modelo efectivo obtenido mediante homogeneización periódica propuesto por Baffico et Al. (2008) y, extendiéndolo al caso armónico en tiempo, nos encontramos con aproximaciones explícitas de la interacción entre la microgeometría y la macroestructura. Esto nos permite concebir un problema inverso, bajo ciertas hipótesis, para estimar el tamaño de los poros en distintas regiones del espécimen.

 

 

October 11, Sebastián Ossandón (IMA-PUCV, Chile)

Modelación de la Corrosión Atmosférica de Metales y Aleaciones Utilizando Redes Neuronales

Abstract: En este trabajo se presentan modelos de entrada-salida para estimar y predecir la velocidad de corrosión del acero y del cobre en diferentes estaciones de experimentación. Los modelos desarrollados se basan en el entrenamiento de una Red Neuronal Artificial (RNA) de base radial para ajustar de manera precisa los datos conocidos u observados. Una vez que se han ajustado los valores de los pesos, en base al entrenamiento antes mencionado, es posible estimar y/o predecir valores para la velocidad de corrosión del acero y del cobre en función de las variables de entrada.

 

 

September 27, Loredana Smaranda (Universidad de Pitesti, Rumania)

Convergence of the Lagrange-Galerkin method for fluid-structure interaction problems

We focus on a numerical method for the discretization of an initial and boundary value problem that models the self–propelled motion of one deformable solid in a bidimensional viscous incompressible fluid. In the model, we suppose that the solid is subjected to a known deformation field representing the action of the aquatic organism muscles. The governing equations consist of the Navier-Stokes equations for the fluid, coupled to Newton’s laws for the solid. The numerical method we propose is based on a global weak formulation, where the nonlinear term in the Navier–Stokes model is discretized using the characteristic function. Since the formulation is global in space, this characteristic function is extended in an appropriated manner inside of the creature, taking into account its deformation. In this talk, we concentrate our attention in the semi-discretization in time and we prove the stability and the convergence of the numerical scheme. The numerical method is consistent enough with the motion of the creature and for this reason, the discretization in space variable is successfully implemented using finite element method.

Abstract_LoredanaSmaranda

 

 

September 13, David Pardo (Universidad del País Vasco & Ikerbasque)

FAST INVERSION OF LOGGING-WHILE-DRILLING (LWD) RESISTIVITY MEASUREMENTS FOR THE CHARACTERIZATION OF HYDROCARBON-BEARING FORMATIONS

A number of three-dimensional (3D) simulators of borehole logging measurements have been developed during the last two decades for oil-industry applications. These simulators have been successfully used to study and quantify different physical effects occurring in 3D geometries. Despite such recent advances, there are still many 3D effects for which reliable simulations are not available. Furthermore, in most of the existing results only partial validations have been reported, typically obtained by comparing solutions of simplified model problems against the corresponding solutions calculated with a lower dimensional (2D

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