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Adaptive finite element methods for parabolic problems. I: A linear model problem. (English) Zbl 0732.65093
An adaptive finite element method for linear parabolic problems is developed. The finite element method uses a space discretization with meshsize variable in space and time and a third-order accurate time discretization with timesteps variable in time.
The algorithm is proven to be: (1) reliable in the sense that the $$L_ 2$$-error in space is guaranteed to be below a given tolerance for all timesteps and (2) efficient in the sense that the approximation error is for the most timesteps not essentially below the given tolerance. Analogous results are given for the corresponding stationary (elliptic) problems.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods 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 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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