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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....

Do Demographic and Disease Structures Affect the Recurrence of Epidemics ?

A. Castellazzo, A. Mauro, C. Volpe, E. Venturino (2012)

Mathematical Modelling of Natural Phenomena

In this paper we present an epidemic model affecting an age-structured population. We show by numerical simulations that this demographic structure can induce persistent oscillations in the epidemic. The model is then extended to encompass a stage-structured disease within an age-dependent population. In this case as well, persistent oscillations are observed in the infected as well as in the whole population.

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 and scientific computing applications

Luca F. Pavarino (2005)

Bollettino dell'Unione Matematica Italiana

This paper reviews the basic mathematical ideas and convergence analysis of domain decomposition methods. These are parallel and scalable iterative methods for the efficient numerical solution of partial differential equations. Two examples are then presented showing the application of domain decomposition methods to large-scale numerical simulations in computational mechanics and electrocardiology.

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...

Domain decomposition methods for solving the Burgers equation

Robert Cimrman (1999)

Applications of Mathematics

This article presents some results of numerical tests of solving the two-dimensional non-linear unsteady viscous Burgers equation. We have compared the known convergence and parallel performance properties of the additive Schwarz domain decomposition method with or without a coarse grid for the model Poisson problem with those obtained by experiments for the Burgers problem.

Domain optimization in 3 D -axisymmetric elliptic problems by dual finite element method

Ivan Hlaváček (1990)

Aplikace matematiky

An axisymmetric second order elliptic problem with mixed boundary conditions is considered. The shape of the domain has to be found so as to minimize a cost functional, which is given in terms of the cogradient of the solution. A new dual finite element method is used for approximate solutions. The existence of an optimal domain is proven and a convergence analysis presented.

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