Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations

Abstract

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.

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

“…Now, arguing as in [21, Theorems 3.1 and 4.1] for n = 2 and [20] for n = 3, we deduce the existence of a positive constant, let us call it µc, that depends on the data on the problem but not on the discretization parameters such that…”

“…This corresponds to an implicit Euler scheme. It is proved in [18,20] that there exist h 0 > 0 and τ 0 > 0 such that…”

“…where we can use the finite element error estimates in [18,20] thanks to the stability estimate in Lemma 6.2 and (43) thanks to the regularity of the optimal control stated in Theorem 3.2.…”

On a Randomized Backward Euler Method for Nonlinear Evolution Equations with Time-Irregular Coefficients

“…Further results on optimal rates under minimal regularity assumptions for linear parabolic PDEs can be found, e.g., in [5,9,21]. For semilinear parabolic problems optimal error estimates are also found in [33], where a discontinuous Galerkin method in time and space is considered. This paper is organized as follows.…”

“…It is also the state-of-the-art method in many recent papers for the numerical solution of evolution equations of the form (1.2). For example, we refer to [5,15,25,33].…”