An LBB‐stable P1/RNP0 finite element based on a pseudo‐random integration method for incompressible and nearly incompressible material flows

Abstract

The aim of this work is to propose a new nodal treatment of the pressure for tetrahedral or triangular meshes devoted to the simulation of incompressible and nearly incompressible material flows. The approach proposed has the interest of fulfilling numerically the LBB condition for P1‐type discretizations over a wide range of element sizes. Thus, the existence of an error estimate is ensured and there is no need for a stabilization technique. For more convenience, the new P1/RNP0 formulation is first detailed for the Stokes problem. It is based on a RNP0 (Random Nodal P0) approximation of the pressure with constant values on nodal subcells whose size is defined by means of a pseudo‐random number generator. For nearly incompressible material flows, the numerical approach is extended to problems involving von Mises elasto‐plasticity. Examples are presented to show the relevance of the new approach for Eulerian and Lagrangian formalisms.