The Mixed Virtual Element Method for the Richards Equation

Abstract

The time-dependent Richards equation can be reformulated as a nonlinear, possibly degenerate, parabolic problem in mixed form by applying the Kirchhoff transformation. A preliminary time integration yields the variational formulation. A numerical treatment of this problem using polygonal and polyhedral meshes is, then, feasible by applying the mixed virtual element method. In this setting, we study a semi-discrete and a fully-discrete virtual element approximation. The theoretical analysis shows that our virtual element formulations are well-posed and convergent, and optimal convergence rates for the approximation errors can be proved. Such theoretical results are confirmed and the accuracy is assessed by investigating the behavior of the method on a set of representative numerical experiments.KeywordsRichard equationMixed virtual element methodPolygonal meshLow-order approximationConvergence analysis