30 research outputs found
On variational eigenvalue approximation of semidefinite operators
Eigenvalue problems for semidefinite operators with infinite dimensional
kernels appear for instance in electromagnetics. Variational discretizations
with edge elements have long been analyzed in terms of a discrete compactness
property. As an alternative, we show here how the abstract theory can be
developed in terms of a geometric property called the vanishing gap condition.
This condition is shown to be equivalent to eigenvalue convergence and
intermediate between two different discrete variants of Friedrichs estimates.
Next we turn to a more practical means of checking these properties. We
introduce a notion of compatible operator and show how the previous conditions
are equivalent to the existence of such operators with various convergence
properties. In particular the vanishing gap condition is shown to be equivalent
to the existence of compatible operators satisfying an Aubin-Nitsche estimate.
Finally we give examples demonstrating that the implications not shown to be
equivalences, indeed are not.Comment: 26 page
Natural hp-BEM for the electric field integral equation with singular solutions
We apply the hp-version of the boundary element method (BEM) for the
numerical solution of the electric field integral equation (EFIE) on a
Lipschitz polyhedral surface G. The underlying meshes are supposed to be
quasi-uniform triangulations of G, and the approximations are based on either
Raviart-Thomas or Brezzi-Douglas-Marini families of surface elements.
Non-smoothness of G leads to singularities in the solution of the EFIE,
severely affecting convergence rates of the BEM. However, the singular
behaviour of the solution can be explicitly specified using a finite set of
power functions (vertex-, edge-, and vertex-edge singularities). In this paper
we use this fact to perform an a priori error analysis of the hp-BEM on
quasi-uniform meshes. We prove precise error estimates in terms of the
polynomial degree p, the mesh size h, and the singularity exponents.Comment: 17 page
Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations
We analyze the conforming approximation of the time-harmonic Maxwell's
equations using N\'ed\'elec (edge) finite elements. We prove that the
approximation is asymptotically optimal, i.e., the approximation error in the
energy norm is bounded by the best-approximation error times a constant that
tends to one as the mesh is refined and/or the polynomial degree is increased.
Moreover, under the same conditions on the mesh and/or the polynomial degree,
we establish discrete inf-sup stability with a constant that corresponds to the
continuous constant up to a factor of two at most. Our proofs apply under
minimal regularity assumptions on the exact solution, so that general domains,
material coefficients, and right-hand sides are allowed
The BEM with graded meshes for the electric field integral equation on polyhedral surfaces
We consider the variational formulation of the electric field integral
equation on a Lipschitz polyhedral surface . We study the Galerkin
boundary element discretisations based on the lowest-order Raviart-Thomas
surface elements on a sequence of anisotropic meshes algebraically graded
towards the edges of . We establish quasi-optimal convergence of
Galerkin solutions under a mild restriction on the strength of grading. The key
ingredient of our convergence analysis are new componentwise stability
properties of the Raviart-Thomas interpolant on anisotropic elements
Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term
We study the mixed formulation of the stochastic Hodge-Laplace problem defined on an n-dimensional domain D (n≥1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (m≥1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element space