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Composite grid finite element method: Implementation and iterative solution with inexact subproblems

Radim Blaheta, P. Byczanski, Roman Kohut (2002)

Applications of Mathematics

This paper concerns the composite grid finite element (FE) method for solving boundary value problems in the cases which require local grid refinement for enhancing the approximating properties of the corresponding FE space. A special interest is given to iterative methods based on natural decomposition of the space of unknowns and to the implementation of both the composite grid FEM and the iterative procedures for its solution. The implementation is important for gaining all benefits of the described...

Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes

Thomas Apel, Anna-Margarete Sändig, Sergey I. Solov'ev (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear...

Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

Thomas Apel, Anna-Margarete Sändig, Sergey I. Solov'ev (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear...

Computing guided modes for an unbounded stratified medium in integrated optics

Fabrice Mahé (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a finite element method to compute guided modes in a stratified medium. The major difficulty to overcome is related to the unboundedness of the stratified medium. Our method is an alternative to the use of artificial boundary conditions and to the use of integral representation formulae. The domain is bounded in such a way we can write the solution on its lateral boundaries in terms of Fourier series. The series is then truncated for the computations over the bounded domain. The problem...

Computing guided modes for an unbounded stratified medium in integrated optics

Fabrice Mahé (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a finite element method to compute guided modes in a stratified medium. The major difficulty to overcome is related to the unboundedness of the stratified medium. Our method is an alternative to the use of artificial boundary conditions and to the use of integral representation formulae. The domain is bounded in such a way we can write the solution on its lateral boundaries in terms of Fourier series. The series is then truncated for the computations over the bounded domain. The problem...

Concepts—An object-oriented software package for partial differential equations

Philipp Frauenfelder, Christian Lage (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such...

Concepts—An Object-Oriented Software Package for Partial Differential Equations

Philipp Frauenfelder, Christian Lage (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into ...

Contact between elastic bodies. II. Finite element analysis

Jaroslav Haslinger, Ivan Hlaváček (1981)

Aplikace matematiky

The paper deals with the approximation of contact problems of two elastic bodies by finite element method. Using piecewise linear finite elements, some error estimates are derived, assuming that the exact solution is sufficiently smooth. If the solution is not regular, the convergence itself is proven. This analysis is given for two types of contact problems: with a bounded contact zone and with enlarging contact zone.

Contact between elastic bodies. III. Dual finite element analysis

Jaroslav Haslinger, Ivan Hlaváček (1981)

Aplikace matematiky

The problem of a unilateral contact between elastic bodies with an apriori bounded contact zone is formulated in terms of stresses via the principle of complementary energy. Approximations are defined by means of self-equilibriated triangular block-elements and an L 2 -error estimate is proven provided the exact solution is regular enough.

Continuous-time finite element analysis of multiphase flow in groundwater hydrology

Zhangxin Chen, Magne Espedal, Richard E. Ewing (1995)

Applications of Mathematics

A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis...

Convergence analysis for an exponentially fitted Finite Volume Method

Reiner Vanselow (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper is devoted to the convergence analysis of a well-known cell-centered Finite Volume Method (FVM) for a convection-diffusion problem in ℝ 2 . This FVM is based on Voronoi boxes and exponential fitting. To prove the convergence of the FVM, we use a new nonconforming Petrov-Galerkin Finite Element Method (FEM) for which the system of linear equations coincides completely with that of the FVM. Thus, by proving convergence properties of the FEM we obtain similar ones for the FVM. For the error...

Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Robert Eymard, Raphaèle Herbin, Jean-Claude Latché, Bruno Piar (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup...

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