Erell Jamelot

Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements

A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with the help of a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, their method required a parameterization of the variational formulation. In order to avoid this difficulty, we use a mixed variational setting instead of the parameterization, which allows us to handle the divergence-free constraint on the field in a straightforward manner. The numerical analysis of the method is carried out, and numerical examples are provided to show the efficiency of our approach.

Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries

A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D geometries with the help of a continuous approximation of the electromagnetic field. In this paper, we investigate how their framework can be adapted to compute the solution to the time-dependent Maxwell equations. In addition, we propose some extensions, such as the introduction of a mixed variational setting and its discretization, to handle the constraint on the divergence of the field.

Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation

Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method for the mixed neutron diffusion equations, discretized with Raviart-Thomas-Nedelec finite elements. This method is based on the Schwarz iterative algorithm with Robin interface conditions to handle communications. We analyse this method from the continuous point of view to the discrete point of view, and we give some numerical results in a realistic highly heterogeneous 3D configuration. Computations are carried out with the MINOS solver of the APOLLO3(R) neutronics code.

Continuous Galerkin Methods for Solving Maxwell Equations in 3D Geometries

Maxwell equations are easily resolved when the computational domain is convex or with a smooth boundary, but if on the contrary it includes geometrical singularities, the electromagnetic field is locally unbounded and globally hard to compute. The challenge is to find out numerical methods which can capture the EM field accurately. Numerically speaking, it is advised, while solving the coupled Maxwell-Vlasov system, to compute a continuous approximation of the field. However, if the domain contains geometrical singularities, continuous finite elements span a strict subset of all possible fields, which is made of the H-regular fields. In order to recover the total field, one can use additional ansatz functions or introduce a weight. The first method, known as the singular complement method [4, 3, 14, 2, 9, 15, 16] works well in 2D and 2D 1 2 geometries and the second method, known as the weighted regularization method [13] works in 2D and 3D. In this contribution, we examine some recent developments of the latter method to solve instationary Maxwell equations and we provide numerical results.

Domain decomposition methods for the diffusion equation with low-regularity solution

We analyze matching and non-matching domain decomposition methods for the numerical approximation of the mixed diffusion equations. Special attention is paid to 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. The domain decomposition method can be non-matching in the sense that the traces of the finite elements spaces may not fit at the interface between subdomains. We prove well-posedness of the discrete problem, that is solvability of the corresponding linear system, provided two algebraic conditions are fulfilled. If moreover the conditions hold independently of the discretization, convergence is ensured.

Domain Decomposition for the Neutron SPN Equations

Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method applied to the neutron SP N equations, which are an approximation of the neutron transport equation. This method is based on the Schwarz iterative algorithm with optimized Robin interface conditions to handle communications. From a computational point of view, this method is rather easy to implement. We give some numerical results in highly heterogeneous 3D configurations. Computations are carried out with the MINOS solver, which is a multigroup SP N solver of the APOLLO3Ⓡ (APOLLO3 is a trademark registered in France.) neutronics code. Numerical experiments show that the method is robust and efficient, and that our choice of the Robin parameters is satisfactory.