Error analysis of mixed finite element methods for nonlinear parabolic equations

TL;DR

This paper establishes a discrete embedding inequality for Raviart--Thomas mixed finite element methods and uses it to derive optimal error estimates for linearized methods applied to nonlinear parabolic equations, supported by numerical validation.

Contribution

It introduces a novel discrete embedding inequality for mixed finite element methods and applies it to obtain optimal error bounds for nonlinear parabolic problems.

Findings

Discrete embedding inequality proved for Raviart--Thomas elements

Optimal error estimates achieved for linearized mixed methods

Numerical examples confirm theoretical results

Abstract

In this paper, we prove a discrete embedding inequality for the Raviart--Thomas mixed finite element methods for second order elliptic equations, which is analogous to the Sobolev embedding inequality in the continuous setting. Then, by using the proved discrete embedding inequality, we provide an optimal error estimate for linearized mixed finite element methods for nonlinear parabolic equations. Several numerical examples are provided to confirm the theoretical analysis.