64 research outputs found
Accurate gradient computations at interfaces using finite element methods
New finite element methods are proposed for elliptic interface problems in
one and two dimensions. The main motivation is not only to get an accurate
solution but also an accurate first order derivative at the interface (from
each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}.
For 2D interface problems, the idea is to introduce a small tube near the
interface and introduce the gradient as part of unknowns, which is similar to a
mixed finite element method, except only at the interface. Thus the
computational cost is just slightly higher than the standard finite element
method. We present rigorous one dimensional analysis, which show second order
convergence order for both of the solution and the gradient in 1D. For two
dimensional problems, we present numerical results and observe second order
convergence for the solution, and super-convergence for the gradient at the
interface
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
A first order system least squares method for the Helmholtz equation
We present a first order system least squares (FOSLS) method for the
Helmholtz equation at high wave number k, which always deduces Hermitian
positive definite algebraic system. By utilizing a non-trivial solution
decomposition to the dual FOSLS problem which is quite different from that of
standard finite element method, we give error analysis to the hp-version of the
FOSLS method where the dependence on the mesh size h, the approximation order
p, and the wave number k is given explicitly. In particular, under some
assumption of the boundary of the domain, the L2 norm error estimate of the
scalar solution from the FOSLS method is shown to be quasi optimal under the
condition that kh/p is sufficiently small and the polynomial degree p is at
least O(\log k). Numerical experiments are given to verify the theoretical
results
Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation
The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems
Computational Engineering
This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)
The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy.
The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism
Breaking spaces and forms for the DPG method and applications including Maxwell equations
Discontinuous Petrov Galerkin (DPG) methods are made easily implementable
using `broken' test spaces, i.e., spaces of functions with no continuity
constraints across mesh element interfaces. Broken spaces derivable from a
standard exact sequence of first order (unbroken) Sobolev spaces are of
particular interest. A characterization of interface spaces that connect the
broken spaces to their unbroken counterparts is provided. Stability of certain
formulations using the broken spaces can be derived from the stability of
analogues that use unbroken spaces. This technique is used to provide a
complete error analysis of DPG methods for Maxwell equations with perfect
electric boundary conditions. The technique also permits considerable
simplifications of previous analyses of DPG methods for other equations.
Reliability and efficiency estimates for an error indicator also follow.
Finally, the equivalence of stability for various formulations of the same
Maxwell problem is proved, including the strong form, the ultraweak form, and a
spectrum of forms in between
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