A priori error estimates of two fully discrete coupled schemes for Biot's consolidation model

TL;DR

This paper provides a priori error estimates for two fully discrete coupled schemes solving Biot's consolidation model, demonstrating their unconditional convergence and optimal error orders through theoretical analysis and numerical validation.

Contribution

It introduces and analyzes two fully discrete schemes for Biot's model using Taylor-Hood elements and combined time discretization methods, establishing their convergence and error bounds.

Findings

Both schemes are unconditionally convergent.

The schemes achieve optimal error orders.

Numerical experiments confirm theoretical results.

Abstract

This paper concentrates on a priori error estimates of two fully discrete coupled schemes for Biot's consolidation model based on the three-field formulation introduced by Oyarzua et al. (SIAM Journal on Numerical Analysis, 2016). The spatial discretizations are based on the Taylor-Hood finite elements combined with Lagrange elements for the three primary variables. For time discretization, we consider two methods. One uses the backward Euler method, and the other applies a combination of the backward Euler and Crank-Nicolson methods. A priori error estimates show that the two schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.