Luc Paquet

A Posteriori error estimation for the dual mixed finite element method of the stokes problem

The paper presents an a posteriori error estimator of residual type for the stationary Stokes problem in a possibly multiply-connected polygonal domain of the plane, using the dual mixed finite element method (FEM). We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution.

Integral Equations for Elliptic Problems with Edge Singularities and Applications to the Fourier-Boundary Element Method

-Let u(x 1,x 2 Z) = be the tangentially regular solution of a transmission problem for the Laplace operator in the horizontal strip × (0,π), the interface being lateral faces of a right prism of height π. We show that each Fourier coefficient ul (x 1,x 2) is the single-layer potential Vl (x 1,x 2 ql ) relative to the two-dimensional Helmholtz operator −δ + l 1. The density ql solves a boundary integral equation, which is uniformly (with respect to the parameter l) well-posed in suitable Sobolev spaces. Decomposition of ql into regular part and singular functions is investigated and used to design an optimally convergent discrete solution ql, h of a boundary element method (BEM) the mesh size of which is explicitly graded and adapted. On the other hand, global regularity of ql in suitable weighted Sobolev spaces is established. This is used to implement a more general optimally convergent BEM with solution ql, h and with mesh size refined only near the corners of the interface. The truncated Fourier series , is then a fully discrete solution of the transmission problem with optimal rate of convergence.

Regularity of the solutions of the steady‐state Boussinesq equations with thermocapillarity effects on the surface of the liquid

In this paper we show that every variational solution of the steady-state Boussinesq equations (u, p, θ) with thermocapillarity effect on the surface of the liquid has the following regularity: u ∈ H2(Ω)2, p ∈ H1(Ω), θ ∈ H2(Ω) under appropriate hypotheses on the angles of the ‘2-D’ container (a cross-section of the 3-D container in fact) and of the horizontal surface of the liquid with the inner surface of the container. The difficulty comes from the boundary condition on the surface of the liquid (e.g. water) which modelizes the thermocapillarity effect on the surface of the liquid (equation (68.10) of Levich [7]). More precisely we will show that u ∈ P22(Ω)2 and that θ ∈ P22(Ω), where P22(Ω) denotes the usual Kondratiev space. This result will be used in a forthcoming paper to prove convergence results for finite element methods intended to compute approximations of a non-singular solution [1] of this problem. Copyright

Refined mixed finite element method for the Boussinesq equations in polygonal domains

Abstract This paper deals with the mixed formulation of the Boussinesq equations in two-dimensional polygonal domains and its numerical approximation. The solutions have a singular behaviour near the non-convex corner points so that we show that they belong to appropriate weighted Sobolev spaces. We further derive appropriate refinement rules on the meshes near the corner points in order to restore the quasi-optimal rate of convergence.