This work is concerned with the numerical solution of the time-harmonic Maxwell equations discretized by discontinuous Galerkin methods on unstructured meshes. Our motivation for using a discontinuous Galerkin method is the enhanced flexibility compared to the conforming edge element method [12]: for instance, dealing with non-conforming meshes is straightforward and the choice of the local app...

Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Ga...

This paper derives upper and lower bounds for the -condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize h are obtained. The theoretical results are applied to finite element approximations of elliptic PDE’s in variational and in mixed form, and to first-order PDE’s approximated usin...

In this work we consider quasi-optimal versions of the Stochastic Galerkin Method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane C . We show that a quasi-optimal approximation is given by...

We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation ∂ t u − ∆u = 0 (0 < α < 1) with initial condition u(x, 0) = v(x) and a homogeneous Dirichlet boundary condition in a bounded polygonal domain Ω. We shall study two semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin FEM, by using piecewise linear functions. We establ...

We give a convergence estimate for a Petrov-Galerkin Algebraic Multigrid method. In this method, the prolongations are defined using the concept of smoothed aggregation while the restrictions are simple aggregation operators. The analysis is carried out by showing that these methods can be interpreted as variational Ritz-Galerkin ones using modified transfer and smoothing operators. The estimat...

Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [SIAM J. Numer. Anal., 47 (2009), pp. 3820–3848] for the acoustic wave equation. Two types of coarse pro...

In this paper we retail the approximation of the clamped plate problem by means of two boundary element methods. In both cases, the variational formulation is given on product of Sobolev spaces and we avoid the orthogonality to polynomials of degree one. The use of biorthogonal wavelets on these spaces leads to a well-conditioned sti ness matrix and reduces strongly the complexity thanks to a c...

— We study the convergence rate of solutions of spectral semi-Galerkin approximations for the équations for the motion of a nonhomogeneous incompressible asymmetrie fluid in a bounded domain. We find error estimâtes that are optimal in the H -norm as well as improved Résumé. — On étudie le taux de convergence d'une approximation de type semi-Galerkin spectrale vers la solution des équations du ...

In this note we consider residual-based a posteriori error estimation for finite element approximations of the transport equation. For the discretization we use piecewise affine continuous or discontinuous finite elements and symmetric stabilization of interior penalty type. The lowest order discontinuous Galerkin method using piecewise constant approximation is included as a special case. The ...