Optimal Error Estimates for Fully Discrete Galerkin Approximations of Semilinear Parabolic Equations
Provides rigorous error analysis for numerical solutions of semilinear parabolic PDEs without growth restrictions, benefiting computational mathematicians.
Proved uniform boundedness of discrete solutions for semilinear parabolic equations with general nonlinearities, enabling optimal error estimates for a dG(0)-finite element scheme.
We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme) and with conforming finite elements in space. The main contribution of this paper is the proof of the uniform boundedness of the discrete solution. This allows us to obtain optimal error estimates with respect to various norms.