A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank--Nicolson, and discontinuous Galerkin $dG(r)$ methods are addressed. For their full discretizations, we employ elliptic reconstructions that are, respectively, piecewise-constant, piecewise-linear, and piecewise-quadratic for $r=1$ in time. We also use certain bounds for the Green's function of the parabolic operator.

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Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions

Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backward Euler and of the Crank-Nicolson methods are considered. For their full discretizations, we use elliptic reconstructions that are, respectively, piecewise-constant and piecewise-linear in time. Certain bounds for the Green’s function of the parabolic operator are also employed.

Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations

FEM discretizations of arbitrary order r are considered for a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers. A posteriori error bounds of interpolation type are derived in the maximum norm. An adaptive algorithm is devised to resolve the boundary layers. Numerical experiments complement our theoretical results.

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A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems

Singularly perturbed boundary value problems are widely studied in applied problems of physics and engineering. However, their solutions are rarely possible to construct in an explicit form, so numerical methods of solving such problems are actively studied. Functions that are the explicit or approximate solution of this problem have huge boundary layer components; therefore, the application of classical interpolation methods leads to significant errors. This paper considers a piecewise-uniform Shishkin mesh, which allows improving the quality of approximation in the boundary layer. A local approximation scheme is implemented, minimal splines are used as basis functions, and the coefficients are calculated as de Boor-Fix type functional values, which are biorthogonal to minimal splines. The results of numerical experiments are presented. They show that the discussed approximation method allows getting accurate approximations of functions that are the solutions of singularly perturbed boundary value problems, in comparison with previously published works.

On biorthogonal approximation of solutions of some boundary value problems on Shishkin mesh

Applications of Mathematics

A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P 1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered. Certain critical details of the implementation are addressed. Based on numerical results we discuss various aspects of the error estimators in particular their effectiveness.

Torsten Linβ

Papers

Numerical Study of Maximum Norm a Posteriori Error Estimates for Singularly Perturbed Parabolic Problems