Monique Dauge

Weighted regularization of Maxwell equations in polyhedral domains

Summary. We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient.

Stationary stokes and Navier-Stokes systems on two-or three-dimensional domains with corners. Part I: linearized equations

The

Neumann and mixed problems on curvilinear polyhedra

H^s

Maxwell and Lamé eigenvalues on polyhedra

-regularity (s being real and nonnegative) of solutions of the Stokes system in domains with corners is studied. In particular, a

Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains

H^2

EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

-regularity result on a convex polyhedron that generalizes Kellogg and Osborn’s result on a convex polygon to three-dimensional domains is stated. Sharper regularity on a cube and on other domains with corners is attained. Conditions for the problem to be Fredholm are also given, and its singular functions along with those of the nonlinear problem are studied in the second part of this paper.

Computation of resonance frequencies for Maxwell equations in non-smooth domains

We prove regularity results inLp Sobolev spaces. On one hand, we state some abstract results byLp functional techniques: exponentially decreasing estimates in dyadic partitions of cones and dihedra, operator valued symbols and Marcinkieviczs theorem. On the other hand, we derive more concrete statements with the help of estimates about the first non-zero eigenvalue of some Laplace-Beltrami operators on spherical domains.

Crack Singularities for General Elliptic Systems

In a convex polyhedron, a part of the Lame eigenvalues with hard simple support boundary conditions does not depend on the Lame coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lame coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non-convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non-H2 singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non-convex polyhedron, the spectrum cannot be approximated by finite element methods using H1 elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains. Copyright

Asymptotics Without Logarithmic Terms for Crack Problems

In the two first parts of this work [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 27e52], [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 343–367] formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated. Now, for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains, the solution of such problems by the finite element method is considered. In order to approximate the solution or the coefficients, different methods are used based on the expressions of the coefficients that were obtained in the first two parts; the dual singular function method is also generalized.

General edge asymptotics of solutions of second-order elliptic boundary value problems I

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwells equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h version of finite elements. We prove exponential convergence of hp FEM for the weighted regularization of Maxwells equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.