A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

Abstract

Abstract.A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement.1. Introduction. The simulation of electromagnetic phenomena with increasing complexity demands accurate and efficient numerical methods suitable for large-scale computing. Finite element (FE) methods are commonly used in this context because they can easily handle complicated geometries by using unstructured grids, provide a rigorous mathematical framework and allow adaptation.In many applications of current interest, the electromagnetic problem is coupled to other physical processes. Salient examples of multiphysics phenomena that include electromagnetics are magnetohydrodynamics (MHD) and plasma physics. These two problems have experienced increasing attention due to the need to develop numerical laboratories in fusion technology design. The simulation of these problems (and many others) would benefit of an all-purpose FE method that would be suitable for the different sub-problems at hand, simplifying the implementation issues and the enforcement of the coupling conditions. In particular, an all-purpose continuous nodal-based formulation would be a favored candidate. E.g. the Navier-Stokes equations are commonly solved with stabilized FE approximations that can deal with the singularly perturbed nature of the system for high Reynolds numbers and circumvent the restrictions related to the corresponding inf-sup condition (see e.g. [13,14]). In plasma physics, fields computed by discontinuous FE Maxwell solvers create a considerable numerical noise when embedded in a plasma code, e.g. using the particle-in-cell method (see [2]). Furthermore, nodal approximations are particularly well-suited for time-dependent electromagnetic problems because the mass matrix can be consistently lumped without loss of accuracy, leading to inexpensive transient solvers.The Maxwell operator has a saddle-point structure, with the particularity that the Lagrange multiplier introduced to enforce the divergence-free constraint is identically zero. Existing FE methods that satisfy the discrete counterpart of the inherent inf-sup condition for this problem are based on Nedelec's or edge elements (see e.g. [27,33]); edge elements lead to fields with discontinuous normal component on element edges or faces. We also refer to alternative formulations based on discontinuous Galerkin approximations [28,24,23,34]. With the aim to solve the Maxwell problem with Lagrangian finite elements (FEs), the differential operator of the problem can be transformed into an elliptic on...

On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

“…The first control is straightforward from (45). It has been shown in [5] that the latter can be obtained if one is able to bound h∇ · u h , properly scaled, in L 2 (Ω). This leads to the introduction of a least squares form of this term in (45).…”

“…This unknown is in fact an artifact to impose condition (37) at the discrete level. See [5,6,4] and references therein for a discussion.…”

Stokes, Maxwell and Darcy: A single finite element approximation for three model problems

“…Thus, nodal interpolations are excluded only when the curl-div form of the problem (12) is being approximated (see [3]). …”

“…Then, a stabilized FE technique for the dual Darcy problem that exhibits the same convergence rates as inf-sup stable FEs was proposed in [4,5]. Very recently, a stabilized FE formulation for the Maxwell problems that allows to use Lagrange finite element methods and converge also to singular solutions has been designed in [3]. Using the stabilized FE approach, all the unknowns for all these problems can be approximated via Lagrangian (nodal) FE spaces.…”

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

“…Although for many years there have been many attempts on how to use the H 1 -conforming finite element method to correctly approximate a non-H 1 -space solution, it is only during the last decade that we have seen a few successful methods. For (1.2), there are the weighted method [31,21,48,25], the H −α -method for some 1/2 < α ≤ 1 [18,11,15], and the L 2 projection method [36,35]. We should also mention the earlier work on the singular complement method [5,6,7,8,9], which deals with Maxwell equations in a domain with reentrant corners as well.…”