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Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H(curl;ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method.

Abstract: We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderón projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new method.

Abstract The finite mass method is a purely Lagrangian scheme for the spatial discretisation of the macroscopic phenomenological laws that govern the flow of compressible fluids. In this article we investigate how to take into account long range gravitational forces in the framework of the finite mass method. This is achieved by incorporating an extra discrete potential energy of the gravitational field into the Lagrangian that underlies the finite mass method. The discretisation of the potential is done in an Eulerian fashion and employs an adaptive tensor product mesh fixed in space, hence the name finite mass mesh method for the new scheme. The transfer of information between the mass packets of the finite mass method and the discrete potential equation relies on numerical quadrature, for which different strategies will be proposed. The performance of the extended finite mass method for the simulation of two-dimensional gas pillars under self-gravity will be reported.

Summary. Many boundary integral equations for exterior boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary G is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an L2(G)-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.

Summary. We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderón projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods.

In papers by Arnold, Falk, and Winther, and by Hiptmair, novel multigrid methods for discrete $\boldsymbol{H}(\textbf{curl};\Omega)$-elliptic boundary value problems have been proposed. Such problems frequently occur in computational electromagnetism, particularly in the context of eddy current simulation. \par This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods. It provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains. In particular, asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems. \par The principal idea is to use approximate Helmholtz-decompositions of the function space $\boldsymbol{H}(\textbf{curl};\Omega)$ into an $H^1(\Omega)$-regular subspace and gradients. The main results of standard multilevel theory for $H^1(\Omega)$-elliptic problems can then be applied to both subspaces. This yields preliminary decompositions still outside the edge element spaces. Judicious alterations can cure this.

Summary.: This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include )- and -conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation operators can be approximated. The stability of -conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation, is investigated both theoretically and in numerical experiments.

The focus of this paper is on boundary value problems for Maxwell's equations that feature cylindrical symmetry both of the domain Ω⊂R$^3$ and the data. Thus, by resorting to cylindrical coordinates, a reduction to two dimensions is possible. However, cylindrical coordinates introduce a potentially malicious singularity at the axis rendering the variational problems degenerate. As a consequence, the analysis of multigrid solvers along the lines of variational multigrid theory confronts severe difficulties. Line relaxation in radial direction and semicoarsening can successfully reign in the degeneracy. In addition, the lack of H$^1$-ellipticity of the double-curl operator entails using special hybrid smoothing procedures. All these techniques combined yield a fast multigrid solver. The theoretical investigation of the method relies on blending generalized Fourier techniques and modern variational multigrid theory. We first determine invariant subspaces of the multigrid iteration operator and analyze the smoothers therein. Under certain assumptions on the material parameters we manage to show uniform convergence of a symmetric V-cycle.

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

Summary.: Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.

We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced. The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semi-coarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at one-dimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid V-cycle are bounded independently of the number of grid levels involved. In addition, the estimates reveal that convergence is also robust with respect to a singular perturbation in one coordinate direction. Finally, we supply numerical evidence that the algorithm performs satisfactorily in settings more general than those covered by the proof.

In the framework of the macroscopic simulation of electrorheological fluids, we present an extension of the classical Bingham model which goes beyond pure shear flows and thus enables the simulation of settings in more complex geometries. Emphasis is on the numerical solution of the resulting nonsmooth minimization problem. We propose the method of augmented Lagrangians combined with an operator-splitting technique which allows to confine the nonlinearity to local, low-dimensional problems. Numerical results are given that illustrate the electrorheological effects for various shear rates and electric field strengths in case of an electrorheological suspension rotating between two revolving cylinders.

We are concerned with the efficient solution of saddle point problems arising from the mixed discretization of 2nd order elliptic problems in two dimensions. We consider the mixed discretization of the boundary value problem by means of lowest order Raviart-Thomas elements. This leads to a saddle point problem, which can be tackled by Uzawa-like iterative solvers. We suggest a prior modification of the saddle point problem according to the augmented Lagrangian approach (cf. [16]) in order to make it more amenable to the iterative procedure. In order to boost the speed of iterative methods, we additionally employ a multilevel preconditioner first presented by Vassilevski and Wang in [26]. It is based on a special splitting of the space of vector valued fluxes, which exploits the close relationship between piecewise linear continuous finite element functions and divergence free fluxes. We prove that this splitting gives rise to an optimal preconditioner: it achieves condition numbers bounded independently on the depth of refinement. The proof is set in the framework of Schwarz methods (cf. [28, 30]). It relies on established results about standard multilevel methods as well as a strengthened Cauchy-Schwarz inequality forRT0-spaces.