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Splitting finite element approximations for quasi-static electroporoelasticity equations

Xuan Liu, Yongkui Zou, Ran Zhang, Yanzhao Cao, Amnon J. Meir

TL;DR

This work addresses the well-posedness and numerical approximation of the quasi-static electroporoelasticity model, which couples Maxwell's equations with Biot's poroelasticity. It develops a splitting backward Euler finite element method that decouples Maxwell-like and poroelastic subproblems, achieving first-order convergence in time and optimal spatial accuracy using Nédélec curl-conforming elements and Lagrange elements. The authors prove existence, uniqueness, and higher-regularity estimates for weak solutions, and derive a rigorous $O(\tau + h)$ error bound for the splitting FE scheme. Numerical experiments confirm stability, accuracy, and substantial computational efficiency gains over classical FEM.

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.

Splitting finite element approximations for quasi-static electroporoelasticity equations

TL;DR

This work addresses the well-posedness and numerical approximation of the quasi-static electroporoelasticity model, which couples Maxwell's equations with Biot's poroelasticity. It develops a splitting backward Euler finite element method that decouples Maxwell-like and poroelastic subproblems, achieving first-order convergence in time and optimal spatial accuracy using Nédélec curl-conforming elements and Lagrange elements. The authors prove existence, uniqueness, and higher-regularity estimates for weak solutions, and derive a rigorous error bound for the splitting FE scheme. Numerical experiments confirm stability, accuracy, and substantial computational efficiency gains over classical FEM.

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.

Paper Structure

This paper contains 14 sections, 18 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Electroporoelasticity equations and weak solution
  3. Uniqueness and existence of weak solution
  4. Uniqueness
  5. Existence
  6. Regularity estimates
  7. Splitting approximation and convergence analysis
  8. Splitting approximate solution
  9. A priori estimates
  10. Convergence order of splitting approximate solution
  11. Finite element discretization and error estimates
  12. Splitting numerical scheme
  13. Error estimates
  14. Numerical experiments

Key Result

lemma thmcounterlemma

For any $p\in H^{1}_{0}$ and $i=1,2,3$, there holds that $\mathcal{B}\frac{\partial}{\partial x_{i}}p = \frac{\partial}{\partial x_{i}}\mathcal{B}p$.

Theorems & Definitions (20)

  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 1
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 2
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • ...and 10 more