We analyse a posteriori error estimates for the discretization with mixed finite elements on simplicial or Cartesian meshes of the multigroup neutron simplified transport (SPN ) equations, in the case where a Robin (or Fourier type) boundary condition is imposed on the boundary. This boundary condition is of particular importance in neutronics, since it corresponds to the well-known vacuum boundary condition. We provide guaranteed and locally efficient estimators. In particular, a specific estimator is designed to handle the Robin boundary condition. We also develop the theory in the case of mixed imposed boundary conditions, of Dirichlet, Neumann or Fourier type. The approach is further extended to a Domain Decomposition Method, the so-called DD+L 2 jumps method. In this framework, the adaptive mesh refinement strategy is implemented for a discretization using Cartesian meshes on each subdomain. Numerical experiments illustrate the theory.

In this paper we introduce Crouzeix-Raviart elements of general polynomial order k and spatial dimension d≥ 2 for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order k is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for k=1 , these degrees of feedom can be split into simplex and ( d-1) dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which correspond to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does not exist in higher space dimension and k>1 .

We are investigating the numerical solution to the 2D time-harmonic Maxwell equations in the presence of a classical medium and a metamaterial, that is with sign-changing coefficients. As soon as the problem has a (unique) solution, we are able to build a converging numerical approximation based on the finite element method, for which there is no constraint on the meshes related to the sign-changing behavior. To that aim, we use Lagrange finite elements to approximate the scalar potentials appearing in the Helmholtz decomposition of the vector-valued electromagnetic fields. Convergence in strong norm is proven for the fields. Numerical examples illustrate the theory.

We solve the Stokes problem numerically. We analyse the P_nc^1× (P^0+P^1) mixed finite element method which exhibits interesting numerical features. However, only an incomplete proof of the inf-sup condition is available. We prove here this condition and the stability of the method.

Using the T-coercivity theory as advocated in Chesnel and Ciarlet (2013) [25], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small.

We analyze aposteriori error estimates for the discretization of the neutron diffusion equations with a Domain Decomposition Method, the so-called DD+L2 jumps method. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. Classically, one introduces a Lagrange multiplier to account for the jumps on the interface. This Lagrange multiplier is used for the reconstruction of the physical variables. Remarkably, no reconstruction of the Lagrange multiplier is needed to achieve the optimal a posteriori estimates.

We consider time-harmonic electromagnetic problems with material coefficients represented by elliptic fields, covering a wide range of complex and anisotropic material media. The properties of elliptic fields are analyzed, with particular emphasis on scalar fields and normal tensor fields. Time-harmonic electromagnetic problems with general elliptic material fields are then studied. Well-posedness results for classical variational formulations with different boundary conditions are reviewed, and hypotheses for the coercivity of the corresponding sesquilinear forms are investigated. Finally, the proposed framework is applied to examples of media used in the literature, such as isotropic lossy media, cloaks of invisibility, magnetized ferrites, and cold plasmas.

There are mainly two techniques that are used to discretize elliptic problems: the finite difference method based upon the strong formulation of the problem (and therefore on classical solutions), and the finite element method based upon the variational formulation (and therefore on weak solutions).

For classical solutions, we saw in the previous chapter that we can go from a problem with second-order derivatives to a variational formulation with just first-order derivatives. By using the generalized integration-by-parts formulae, now we will justify this transition to the lower order derivatives in the general case. Once that is done, we will solve variational formulations that are defined in Sobolev spaces, allowing us to establish the existence and uniqueness of solutions to the Dirichlet, Neumann and Fourier problems. We place ourselves in Ω, an open set of ℝd whose boundary is Lipschitz and bounded.

This chapter is dedicated to the practical aspects used in the finite element method. We start by asking questions about the computer implementation of the method before continuing onto the algorithmic considerations, and then concluding with some numerical illustrations obtained via Matlab. The Matlab scripts presented here are easily translatable into other scripting languages (Python, Julia, etc.).

At the core scale, neutron deterministic calculations are usually based on the neutron diffusion equation. Classically, this equation can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. The goal is to extend the Adaptive Mesh Refinement (AMR) strategy previously proposed in [1] to the Domain Decomposition+L2 jumps which allows non conformity at the interface between subdomains. We are able to refine each subdomain independently, which eventually leads to a more optimal refinement. We numerically investigate the improvements made to the AMR strategy.

The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the -coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.

There are several steady-state physical systems (i.e. the state reached when t → ∞), which are modeled using elliptic partial differential equations.

In this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of T-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on an augmented variational formulation obtained with the help of the weighted regularization method.

Classically, the well-posedness of variational formulations of mixed linear problems is achieved through the inf-sup condition on the constraint. In this note, we propose an alternative framework to study such problems by using the T- coercivity approach to derive a global inf-sup condition. Generally speaking, this is a constructive approach that, in addition, drives the design of suitable approximations. As a matter of fact, the derivation of the uniform discrete inf-sup condition for the approximate problems follows easily from the study of the original problem. To support our view, we solve a series of classical mixed problems with the T- coercivity approach. Among others, the celebrated Fortin Lemma appears naturally in the numerical analysis of the approximate problems.

When finite elements are used to discretize linear elliptic problems, large linear systems arise whose matrix structures are very particular. They are sparse, which is to say that they possess a very large number of zero entries (see theorems 5.2 and 5.3). It is therefore important to understand the main techniques that are used to solve linear systems, so that the most appropriate one can be chosen depending on the nature of the linear system. This appendix is for those who may have questions about how the finite element method can be implemented in Fortran, C/C++, or any other fast programming language. In the programming environments used within scientific computing (Matlab, Scilab, Octave, Python, Julia, etc.), matrix storage issues and system-solving algorithms are obfuscated by the terminology used (sparse-type in Matlab), but understanding these techniques will be very informative.

In this manuscript, we study solutions to resonant Maxwell’s equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω2u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [12] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series of numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.