Fully discrete error estimation by the method of lines for a nonlinear parabolic problem. (English) Zbl 1099.65091

The author considers approximate solution of a nonlinear second-order parabolic problem in one space dimension with homogeneous Dirichlet boundary conditions. The finite space interval is discretized and a hierarchical basis of finite elements with polynomials of the degree \(p\) is employed. A singly implicit Runge-Kutta method is used for the time discretization. A posteriori error estimates are defined by means of “bubble functions” of the degree \(p + 1\).
The author assumes that the norms of the spatial and time discretization are proportional and tend to zero. Then he proves that the ratio of the \(H^{-1}\)-norms of the a posteriori error estimate and the actual error, respectively, tends to one.


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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