IMA Journal of Numerical Analysis - current issue

Convergence of an adaptive FEM for a class of degenerate convex minimization problems

Carstensen, C. — 2008-06-24

A class of degenerate convex minimization problems allows for some adaptive finite-element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of successive loops of the form and employs the bulk criterion. The convergence in L^p^'(; ^m ^x ⁿ) relies on new sharp strict convexity estimates of degenerate convex minimization problems with The class of minimization problems includes strong convex problems and allows applications in an optimal design task, Hencky elastoplasticity or relaxation of two-well problems allowing for microstructures.

Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates

Grote, M. J., Schneebeli, A., Schotzau, D. — 2008-06-24

We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375–386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L²-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetics.

Discretization of coupled heat and electrical diffusion problems by finite-element and finite-volume methods

Bradji, A., Herbin, R. — 2008-06-24

We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which couples the equations. A finite-element scheme and a finite-volume scheme are considered for the discretization of the system; in both cases, we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system.

An optimal L{infty}(L2)-error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection-diffusion problem

Dolejsi, V., Feistauer, M., Kucera, V., Sobotikova, V. — 2008-06-24

This paper is concerned with the analysis of the discontinuous Galerkin finite-element method applied to the space semi-discretization of a nonlinear non-stationary convection–diffusion problem. Attention is paid on the derivation of an L(L²)-optimal error estimate for the symmetric interior penalty Galerkin scheme. The error analysis is performed for standard simplicial meshes under the assumption that the exact solution of the problem and the solution of an elliptic dual problem are sufficiently regular. The theoretical results are illustrated by numerical experiments.

A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems

Freitag, M. A., Spence, A. — 2008-06-24

In this paper, we consider the computation of an eigenvalue and the corresponding eigenvector of a large sparse Hermitian positive-definite matrix using inexact inverse iteration with a fixed shift. For such problems, the large sparse linear systems arising at each iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomplete Cholesky factorization and derive a new tuned Cholesky preconditioner which shows considerable improvement over the standard preconditioner. This improvement is analysed using the convergence theory for MINRES. We also compare the spectral properties of the tuned preconditioned matrix with those of the standard preconditioned matrix. In particular, we provide both a perturbation result and an interlacing result, and these results show that the spectral properties of the tuned preconditioner are similar to those of the standard preconditioner. For Rayleigh quotient shifts, comparison is also made with a technique introduced by Simoncini & Eldén (2002, BIT, 42, 159–182) which involves changing the right-hand side of the inverse iteration step. Several numerical examples are given to illustrate the theory described in the paper.

Stability properties of discontinuous Galerkin methods for 2D elliptic problems

Marazzina, D. — 2008-06-24

We address the problem of finding the necessary stabilization for a class of discontinuous Galerkin methods in mixed form for the 2D case. In particular, we present a new stabilized formulation of the (unstable) Bassi–Rebay method and a new formulation of the local discontinuous Galerkin method. The stability properties of the new formulations are studied and error estimates are derived. The theoretical results are validated in a series of numerical tests.

Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence

Sun, W., Wu, J. — 2008-06-24

In this paper, we present a general framework for interpolatory quadrature rules for Hadamard finite-part integrals with a second-order singularity. Gaussian quadrature rules are viewed as a special case and many interesting features can be obtained easily from the framework. We prove theoretically the equivalence of some existing formulas which were obtained in different ways. We show the point-wise superconvergence of these interpolatory quadrature rules, i.e. when the singular point coincides with certain a priori known points, the accuracy is better than what is generally possible. The extension of a popular interpolatory quadrature rule for Cauchy principal value integrals is presented. A new quadrature rule of Gaussian type is proposed for the evaluation of integrals simultaneously involving different types of singularities. Numerical examples confirm our theoretical results

Block preconditioning of real-valued iterative algorithms for complex linear systems

Benzi, M., Bertaccini, D. — 2008-06-24

We revisit real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. We argue that if either the real or the symmetric part of the coefficient matrix is positive semidefinite, block preconditioners for real equivalent formulations may be a useful alternative to preconditioners for the original complex formulation. Numerical experiments illustrating the performance of the various approaches are presented.

Uzawa-based adaptive methods for linear output functionals

Micheletti, S., Perotto, S., Verani, M. — 2008-06-24

In this paper, we address the approximation of a linear output functional J(u) to within a prescribed tolerance , u being the solution of an elliptic problem, via a new iterative procedure named adaptive goal-oriented (ago) algorithm. The core of this scheme is to extend to a goal-oriented setting the good properties of the adaptive Uzawa algorithm proposed in Dahlke et al. (2000, Math. Model. Numer. Anal., 34, 1003–1022), Dahlke et al. (2002, SIAM J. Numer. Anal., 40, 1230–1262) and Bänsch et al. (2002, SIAM J. Numer. Anal., 40, 1207–1229): reliability, automatism and flexibility in the choice of the discrete spaces. A convergence analysis of the ago algorithm is also carried out and some test cases are provided to assess its reliability in the 2D case.