Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

Antoine Tambue,Jean Daniel Mukam

doi:10.1016/j.indag.2020.06.008

Abstract

In this work, numerical approximation of the second order non-autonomous semilinear parabolic partial differential equations (PDEs) is investigated using the classical finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the semi-discrete scheme based on finite element method toward the mild solution of semilinear non-autonomous PDEs under polynomial growth and one-sided Lipschitz conditions of the nonlinear term. Our convergence rate is obtained with general non-smooth initial data, and is similar to that of the autonomous case. Such convergence result is very important in numerical analysis. For instance, it is one step forward for numerical approximation of non-autonomous stochastic partial differential equations with the finite element method.

Highlights

Nonlinear partial differential equations are powerful tools in modelling real-world phenomena in many fields such as in geo-engineering
Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the semi-discrete scheme based on finite element method toward the mild solution of semilinear non-autonomous partial differential equations (PDEs) under polynomial growth and one-sided Lipschitz conditions of the nonlinear term
Our convergence rate is obtained with general non-smooth initial data, and is similar to that of the autonomous case

Summary

Introduction

Nonlinear partial differential equations are powerful tools in modelling real-world phenomena in many fields such as in geo-engineering. Our result is very useful while studying the convergence of the finite element method for many nonlinear problems, including stochastic partial differential equations(SPDEs), see for example [2,3,14] and references therein for time independent SPDEs. in the case of SPDEs, due to the Ito-isometry or the Burkholder Davis–Gundy inequality, the non-smooth version of Lemma 3.1 cannot be applied since it brings degenerated integrals, which cause difficulties in the error estimates or reduce considerably the order of convergence. Establishing Lemma 3.1, which is our key ingredient Such results for autonomous problems with self-adjoint linear operator were thoroughly investigated in [13], revealing the importance of such estimate in numerical analysis. ∥(− A(s)) 2 u(t)∥ + ∥F(u(t))∥ ≤ C 1 + ∥(− A(s)) 2 u0∥ , β ∈ [0, 2), s, t ∈ [0, T ]

Finite element discretization

Error estimate of the semilinear problem under global lipschitz condition