Stabilized mixed continuous/enriched Galerkin formulations for locally mass conservative poromechanics

Abstract

Local (element-wise) mass conservation is often highly desired for numerical solutions to coupled poromechanical problems. As an efficient numerical method featuring this property, mixed continuous Galerkin (CG)/enriched Galerkin (EG) finite elements have recently been proposed in which piecewise constant functions are enriched to the pore pressure interpolation functions of the conventional mixed CG/CG elements. While this enrichment of the pressure space provides local mass conservation, it unavoidably alters the stability condition for mixed finite elements. Because no stabilization method has been available for the new stability condition, high-order displacement interpolation has been required for mixed CG/EG elements if undrained condition is expected. To circumvent this requirement, here we develop stabilized formulations for the mixed CG/EG elements that permit equal-order interpolation functions even in the undrained limit. We begin by identifying the inf–sup condition for mixed CG/EG elements by phrasing an enriched poromechanical problem as a twofold saddle point problem. We then derive two types of stabilized formulations, one based on the polynomial pressure projection (PPP) method and the other based on the fluid pressure Laplacian (FPL) method. A key finding of this work is that both methods lead to stabilization terms that should be augmented only to the CG part of the pore pressure field, not to the enrichment part. The two stabilized formulations are verified and investigated through numerical examples that involve various conditions spanning 1D to 3D, isotropy to anisotropy, and homogeneous to heterogeneous domains. The methodology presented in this work may also help stabilize other types of mixed finite elements in which the constraint field is enriched by additional functions.