Numerical Analysis of a Finite Element / Volume Penalty Method

Abstract. We present here some contributions to the numerical analysis of the penalty method in the finite element context. We are especially interested in the ability provided by this approach to use Cartesian, non boundary-fitted meshes to solve elliptic problems in complicated domain. In the spirit of fictitious domains, the initial problem is replaced by a penalized one, posed over a simply shaped domain which covers the original one. This method relies on two parameters, namely h (space-discretization parameter) and ε (penalty parameter). We propose here a general strategy to estimate the error in both parameters, and we present how it can be applied to various situations. We pay special attention to a scalar version of the rigid motion constraint for fluid-particle flows