Splitting finite element approximations for quasi-static electroporoelasticity equations

TL;DR

This paper develops and analyzes a splitting backward Euler finite element method for electroporoelasticity equations, proving its stability, convergence, and computational efficiency through theoretical analysis and numerical experiments.

Contribution

It introduces a novel splitting finite element scheme for electroporoelasticity equations with rigorous error analysis and demonstrates improved computational efficiency.

Findings

Proved well-posedness and stability of the splitting scheme

Established error estimates for convergence in space and time

Numerical results confirm reduced computational complexity

Abstract

The electroporoelasticity model, which couples Maxwell's equations with Biot's equations, plays a critical role in applications such as water conservancy exploration, earthquake early warning, and various other fields. This work focuses on investigating its well-posedness and analyzing error estimates for a splitting backward Euler finite element method. We first define a weak solution consistent with the finite element framework. Then, we prove the uniqueness and existence of such a solution using the Galerkin method and derive a priori estimates for high-order regularity. Using a splitting technique, we define an approximate splitting solution and analyze its convergence order. Next, we apply Nedelec's curl-conforming finite elements, Lagrange elements, and the backward Euler method to construct a fully discretized scheme. We demonstrate the stability of the splitting numerical solution and provide error estimates for its convergence order in both temporal and spatial variables. Finally, we present numerical experiments to validate the theoretical results, showing that our method significantly reduces computational complexity compared to the classical finite element method.