Adaptive finite element methods for parabolic problems. II: Optimal error estimates in \(L_ \infty L_ 2\) and \(L_ \infty L_ \infty\). (English) Zbl 0830.65094

Eriksson, Kenneth; Johnson, Claes

SIAM J. Numer. Anal. 32, No. 3, 706-740 (1995).

Summary: [For part I see ibid. 28, No. 1, 43-77 (1991; Zbl 0732.65093).]
Optimal error estimates are derived for a complete discretization of linear parabolic problems using space-time finite elements. The discretization is done first in time using the discontinuous Galerkin method and then in space using the standard Galerkin method. The underlying partitions in time and space need not be quasi-uniform and the partition in space may be changed from time step to time step. The error bounds show, in particular, that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same (given) level, i.e., without error accumulation effects.
The derivation of the estimates is based on the orthogonality of the Galerkin procedure and the use of strong stability estimates. The particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration.

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
65M50	Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35K15	Initial value problems for second-order parabolic equations

Keywords:

optimal error estimates; linear parabolic problems; finite elements; discontinuous Galerkin method; stability; adaptive methods; reliable automatic error control

Citations:

Zbl 0732.65093