Parameter Robust Isogeometric Methods for a Four-Field Formulation of Biot's Consolidation Model

TL;DR

This paper introduces a robust isogeometric finite element method for Biot's consolidation model using a four-field formulation, achieving optimal error estimates that are insensitive to material parameters, especially in nearly incompressible cases.

Contribution

The paper develops a novel isogeometric discretization for Biot's model with a four-field formulation, providing the first proof of parameter-robust optimal error estimates without requiring a positive storage coefficient.

Findings

Numerical experiments confirm theoretical error estimates.

High-order convergence rates are achieved.

Method performs well across various mesh sizes and material parameters.

Abstract

In this paper, a novel isogeometric method for Biot's consolidation model is constructed and analyzed, using a four-field formulation where the unknown variables are the solid displacement, solid pressure, fluid flux, and fluid pressure. Mixed isogeometric spaces based on B-splines basis functions are employed in the space discretization, allowing a smooth representation of the problem geometry and solution fields. The main result of the paper is the proof of optimal error estimates that are robust with respect to material parameters for all solution fields, particularly in the case of nearly incompressible materials. The analysis does not require a uniformly positive storage coefficient. The results of numerical experiments in two and three dimensions confirm the theoretical error estimates and high-order convergence rates attained by the proposed isogeometric Biot discretization and assess its performance with respect to the mesh size, spline polynomial degree, spline regularity, and material parameters.