9,605 research outputs found
The hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces: a priori error analysis
This paper presents an a priori error analysis of the hp-version of the
boundary element method for the electric field integral equation on a piecewise
plane (open or closed) Lipschitz surface. We use H(div)-conforming
discretisations with Raviart-Thomas elements on a sequence of quasi-uniform
meshes of triangles and/or parallelograms. Assuming the regularity of the
solution to the electric field integral equation in terms of Sobolev spaces of
tangential vector fields, we prove an a priori error estimate of the method in
the energy norm. This estimate proves the expected rate of convergence with
respect to the mesh parameter h and the polynomial degree p
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
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
On the convergence of the hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces
In this paper the hp-version of the boundary element method is applied to the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. The underlying meshes are supposed to be quasi-uniform. We use \bH(\div)-conforming discretisations with quadrilateral elements of Raviart-Thomas type and establish quasi-optimal convergence of hp-approximations. Main ingredient of our analysis is a new \tilde\bH^{-1/2}(\div)-conforming p-interpolation operator that assumes only \bH^r\cap\tilde\bH^{-1/2}(\div)-regularity () and for which we show quasi-stability with respect to polynomial degrees
hp-version time domain boundary elements for the wave equation on quasi-uniform meshes
Solutions to the wave equation in the exterior of a polyhedral domain or a
screen in exhibit singular behavior from the edges and corners.
We present quasi-optimal -explicit estimates for the approximation of the
Dirichlet and Neumann traces of these solutions for uniform time steps and
(globally) quasi-uniform meshes on the boundary. The results are applied to an
-version of the time domain boundary element method. Numerical examples
confirm the theoretical results for the Dirichlet problem both for screens and
polyhedral domains.Comment: 41 pages, 11 figure
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