Error analysis of a fictitious domain method applied to a Dirichlet problem

Abstract

In this paper, we analyze the error of a fictitious domain method with a Lagrange multiplier. It is applied to solve a non homogeneous elliptic Dirichlet problem with conforming finite elements of degree one on a regular grid. The main point is the proof of a uniform inf-sup condition that holds provided the step size of the mesh on the actual boundary is sufficiently large compared to the size of the interior grid.Dans cet article, nous etudions l'erreur d'une methode de domaine fictif avec multiplicateur de Lagrange. Nous l'appliquons a. la resolution d'un problema elliptique avec condition de Dirichlet non-homogene au bord par une methode d'elements finis conforme de degre un sur une grille uniforme. Ceci repose sur la demonstration d'une condition inf-sup uniforrne qui est satisfaite lorsque Ie pas de la discretisation sur la frontiere du dornaine d'origine est suffisamrnent grand compare au pas de la grille interieure.

A stabilized Lagrange multiplier method for the enriched finite-element approximation of Tresca contact problems of cracked elastic bodies

“…Similarly to the classical result presented in [16], we prove that an inf-sup condition is satisfied between V h and W H for minimal size of 3h for the patches (see Appendix A for the proof on a scalar field, which can be straightforwardly generalized to vector field). This implies in particular that an optimal convergence can be reached if the multiplier is taken in W H .…”

“…We have the length of each segment S of the subdivision S H of Γ C is not less than 3h, then similarly to [16] we can find a node a S such that the macro-element ∆ S consisting of the six triangles of T h with common vertex a S satisfies the following properties:…”

“…Note that for a relatively large mesh size, the local projection stabilization has no influence. Now, concerning the minimal patch size, the inf-sup condition is proven to be satisfied in [16] for a size greater or equal to 3h. Numerically, the inf-sup condition seems to be satisfied for smaller values of the minimal patch size.…”

High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation

“…Difficulties arise when imposing the boundary conditions (especially Dirichlet): Ad-hoc approaches such as Lagrange multipliers or penalty methods have to be considered [4]. Moreover the treatment of material interfaces need special formulations [5,6] if one wants to keep optimal converge properties. Recently, Düster et al [7,8] have proposed a high order extension of the fictitious domain approach coined Finite Cell method.…”

A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails

“…Thus, we will be able to avoid remeshing the fluid domain when the sail shape changes along the iterations. The analysis of this method follows essentially the papers by Glowinski et al [7] and Girault and Glowinski [5]. However, as explained above, we have to adapt these techniques because of the lack of regularity of the solution, due to the singularities on the boundary of the flow domain.…”