Convergence of a Full Discretization of Quasi-linear Parabolic Equations in Isotropic and Anisotropic Orlicz Spaces
Etienne Emmrich, Aneta Wróblewska‐Kamińska · SIAM Journal on Numerical Analysis · 2013
Convergence of a subsequence of approximate solutions arising from a full discretization is shown for a general class of quasi-linear parabolic problems. The numerical method combines the backward Euler method for the time discretization with a generalized internal approximation scheme for the spatial discretization. The governing monotone elliptic differential operator is described by a nonlinearity that may have anisotropic and nonpolynomial growth but fulfills a coercivity condition in terms of a generalized $\mathscr{N}$-function. If the problem admits a unique solution, which is shown in the case of a strictly monotone nonlinearity and in a class of sufficiently smooth solutions, then the whole sequence of approximate solutions converges. Moreover, an a priori error estimate for the temporal semidiscretization is provided and a numerical illustration is given.