Section: New Results

Numerical analysis for PDEs

A family of Crouzeix-Raviart Finite Elements in 3D

Participant : Patrick Ciarlet.

This work is done in collaboration with C. Dunkl (University of Virginia) and S. Sauter (Universität Zürich).

We develop a family of non-conforming “Crouzeix–Raviart” type finite elements in three dimensions. They consist of local polynomials of maximal degree $p$ on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally, we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.

Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

Participants : Patrick Ciarlet, Léandre Giret, Félix Kpadonou.

This work is done in collaboration with E. Jamelot (CEA).

We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method.

Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients

Participant : Patrick Ciarlet.

This work is done in collaboration with M. Vohralik (EPI SERENA).

We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments.

On the convergence in $H^1$-norm for the fractional Laplacian

Participant : Patrick Ciarlet.

This work is done in collaboration with J.P. Borthagaray (University of Maryland).

We consider the numerical solution of the fractional Laplacian of index $s\in (1/2,1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space ${\tilde{H}}^s\left(\Omega \right)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1\left(\Omega \right)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1\left(\Omega \right)$. A natural question is then whether one can obtain error estimates in $H^1\left(\Omega \right)$-norm, in addition to the classical ones that can be derived in the ${\tilde{H}}^s\left(\Omega \right)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.