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We consider the solution of second order elliptic PDEs in Rd with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the Poisson equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H1 / 2–stable projection, for instance, the L2–projection for p = 1 or the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which...
In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the norm for the sequence of discrete operators....
In this paper we consider the Maxwell resolvent operator and its finite element
approximation. In this framework it is natural the use of the edge element
spaces and to impose the divergence constraint in a weak
sense with the introduction of a Lagrange multiplier, following
an idea by Kikuchi [14].
We shall review some of the known properties for edge element
approximations and prove some new result. In particular we shall prove a
uniform convergence in the L2 norm for the sequence of discrete...
We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy...
We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a...
In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter
dependence to problems involving (a) nonaffine dependence on the
parameter, and (b) nonlinear dependence on the field variable.
The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational
decomposition. We first review the coefficient function...
The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable...
The accuracy of the domain embedding method from [A. Rieder, Modél. Math.
Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problems
suffers under a coarse boundary approximation. To overcome this drawback the method
is furnished with
an a priori (static) strategy for an adaptive approximation space refinement near the
boundary. This is done by selecting suitable wavelet subspaces.
Error estimates and
numerical experiments validate the proposed adaptive scheme.
In contrast to similar,...
We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We...
We present a numerical algorithm to
solve the micromagnetic equations based on tangential-plane
minimization for the magnetization update and a homothethic-layer
decomposition of outer space for the computation of the demagnetization field.
As a first application, detailed results on the flower-vortex
transition in the cube of Micromagnetic Standard Problem number 3 are
obtained, which confirm, with a different method, those already
present in the literature, and validate our method and...
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