Ricardo Oyarzúa, Sander Rhebergen, Manuel Solano, Paulo Zuñiga:
Error analysis of a conforming and locking-free four-field formulation in poroelasticity
We present an a priori and a posteriori error analysis of a conforming finite element method for a four-field formulation of Biot's consolidation model. For the a priori error analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart--Thomas elements of order k>=0 for the fluid flux, discontinuous polynomials of degree k for the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-based a posteriori error estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposed a posteriori error indicator.
This preprint gave rise to the following definitive publication(s):
Ricardo OYARZúA, Sander RHEBERGEN, Manuel SOLANO, Paulo ZUñIGA: Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model. ESAIM: Mathematical Modelling and Numerical Analysis, vol 55, pp. S475-S506, (2021).