Gradient superconvergence on uniform simplicial partitions of polytopes. (English) Zbl 1042.65081

Brandts, Jan; Křížek, Michal

IMA J. Numer. Anal. 23, No. 3, 489-505 (2003).

For the solution of the Poisson equation with Dirichlet boundary conditions, the linear simplicial finite-element method is discussed. Superconvergence (supercloseness) of the gradient for this method is proved in a new and elegant manner, independent of the space dimension. As a new result, superconvergence for dimension four and up is proved simultaneously. The authors embed the gradients of the continuous piecewise linear functions into a larger space for which is constucted an orthonormal basis with symmetry properties. As the authors show, the supercloseness of the gradient may be used in the study of more complicated three-dimensional mixed finite-element methods.

Reviewer: Iulian Coroian (Baia Mare)

MSC:

65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

uniform partition; \(n\)-simplex; point symmetry; elliptic problems; finite-element method; superconvergence; Poisson equation