S. Nicaise

Bio: S. Nicaise is an academic researcher. The author has contributed to research in topics: Dirichlet boundary condition & Finite element method. The author has an hindex of 1, co-authored 1 publications receiving 46 citations.

The Nitsche mortar finite‐element method for transmission problems with singularities

Abstract: The paper deals with a Nitsche-type finite-element method for treating non-matching meshes at the interface of some domain decomposition. The method is extended to transmission (or interface) problems of the plane with Dirichlet boundary conditions entailing singularities at the corners or endpoints of the polygonal interface. In a natural way, the interface of the transmission problem is taken as the interface of the domain decomposition and of the non-matching meshes. Properties of the finite-element scheme and error estimates are proved. For appropriate local mesh refinement, optimal convergence rates as known for the classical finite-element method and regular solution are derived. Some numerical tests illustrate the approach and confirm the theoretical results.

A finite element method for domain decomposition with non-matching grids

Abstract: In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity

Abstract: We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection―diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Peclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Peclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

Nitsche's method for interface problems in computational mechanics

Abstract: We give a review of Nitsche’s method applied to interface problems, involving real or artificial interfaces Applications to unfitted meshes, Chimera meshes, cut meshes, fictitious domain methods,

A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

Abstract: In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poisson’s equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection

Abstract: We construct and analyze a discontinuous Galerkin method to solve advection-diffusion-reaction PDEs with anisotropic and semidefinite diffusion. The method is designed to automatically detect the so-called elliptic/hyperbolic interface on fitted meshes. The key idea is to use consistent weighted average and jump operators. Optimal estimates in the broken graph norm are proven. These are consistent with well-known results when the problem is either hyperbolic or uniformly elliptic. The theoretical results are supported by numerical evidence.