A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations
Summary: Dirichlet problem is considered for a singularly perturbed semilinear parabolic convection-diffusion equation on a rectangular domain. The solution of the classical finite difference scheme on a uniform mesh converges at the rate \(\mathcal O((\varepsilon + N^{-1})^{-1} N^{-1} + N_0^{-1})\) where \(N + 1\) and \(N_0 + 1\) denote the numbers of mesh points with respect to \(x\) and \(t\) respectively, \(\varepsilon \in (0,1]\) is the perturbation parameter. Using nonlinear and linearised basic classical schemes, finite difference schemes on a posteriori adaptive meshes based on uniform subgrids are constructed. The subdomains where grid refinement is required are defined by the gradients of the solutions of the intermediate discrete problems. The constructed difference schemes converge 'almost \(\varepsilon \)-uniformly', namely, at the rate \(\mathcal O((\varepsilon ^{-\nu} N^{-1}) + N^{-1/2} + N_0^{-1})\) where \(\nu \) is an arbitrary number from \((0, 1]\).
semilinear parabolic convection-diffusion equations
almost \(\varepsilon \)-uniform convergence
scientific article; zbMATH DE number 5703887