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Discontinuous Galerkin methods for problems with Dirac delta source∗

Paul Houston, Thomas Pascal Wihler (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement ...

Discrete anisotropic curvature flow of graphs

Klaus Deckelnick, Gerhard Dziuk (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfaces in natural norms. Furthermore we prove a global existence result for the continuous problem and present some examples of computed surfaces.

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

Emmanuel Creusé, Serge Nicaise (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions.

Discrete maximum principle for interior penalty discontinuous Galerkin methods

Tamás Horváth, Miklós Mincsovics (2013)

Open Mathematics

A class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of...

Discrete Sobolev inequalities and L p error estimates for finite volume solutions of convection diffusion equations

Yves Coudière, Thierry Gallouët, Raphaèle Herbin (2001)

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

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce L p error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications

Jürgen Geiser (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are developed for solving the arising systems of convection-diffusion-dispersion-reaction equations, and the received results of several discretization methods are presented. We concentrate on linear reaction systems, which can be solved analytically. In the numerical methods, we use large time-steps to achieve long simulation times of about 10 000 years. We propose...

Divergence boundary conditions for vector Helmholtz equations with divergence constraints

Urve Kangro, Roy Nicolaides (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The idea of replacing a divergence constraint by a divergence boundary condition is investigated. The connections between the formulations are considered in detail. It is shown that the most common methods of using divergence boundary conditions do not always work properly. Necessary and sufficient conditions for the equivalence of the formulations are given.

Divergence of FEM: Babuška-Aziz triangulations revisited

Peter Oswald (2015)

Applications of Mathematics

By re-examining the arguments and counterexamples in I. Babuška, A. K. Aziz (1976) concerning the well-known maximum angle condition, we study the convergence behavior of the linear finite element method (FEM) on a family of distorted triangulations of the unit square originally introduced by H. Schwarz in 1880. For a Poisson problem with polynomial solution, we demonstrate arbitrarily slow convergence as well as failure of convergence if the distortion of the triangulations grows sufficiently fast....

Domain decomposition algorithms for time-harmonic Maxwell equations with damping

Ana Alonso Rodriguez, Alberto Valli (2001)

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

Three non-overlapping domain decomposition methods are proposed for the numerical approximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). For each method convergence is proved and, for the discrete problem, the rate of convergence of the iterative algorithm is shown to be independent of the number of degrees of freedom.

Domain Decomposition Algorithms for Time-Harmonic Maxwell Equations with Damping

Ana Alonso Rodriguez, Alberto Valli (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Three non-overlapping domain decomposition methods are proposed for the numerical approximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). For each method convergence is proved and, for the discrete problem, the rate of convergence of the iterative algorithm is shown to be independent of the number of degrees of freedom.

Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions

Ladislav Foltyn, Dalibor Lukáš, Ivo Peterek (2020)

Applications of Mathematics

We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At...

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