8,030 research outputs found
Boundary elements with mesh refinements for the wave equation
The solution of the wave equation in a polyhedral domain in
admits an asymptotic singular expansion in a neighborhood of the corners and
edges. In this article we formulate boundary and screen problems for the wave
equation as equivalent boundary integral equations in time domain, study the
regularity properties of their solutions and the numerical approximation.
Guided by the theory for elliptic equations, graded meshes are shown to recover
the optimal approximation rates known for smooth solutions. Numerical
experiments illustrate the theory for screen problems. In particular, we
discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann
operator and applications to the sound emission of tires.Comment: 45 pages, to appear in Numerische Mathemati
An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations
We propose an adaptive finite element method for the solution of a
coefficient inverse problem of simultaneous reconstruction of the dielectric
permittivity and magnetic permeability functions in the Maxwell's system using
limited boundary observations of the electric field in 3D. We derive a
posteriori error estimates in the Tikhonov functional to be minimized and in
the regularized solution of this functional, as well as formulate corresponding
adaptive algorithm. Our numerical experiments justify the efficiency of our a
posteriori estimates and show significant improvement of the reconstructions
obtained on locally adaptively refined meshes.Comment: Corrected typo
Two fluid space-time discontinuous Galerkin finite element method. Part I: numerical algorithm
A novel numerical method for two fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local {\it hp}-refinement. The front tracking is incorporated via cut-cell mesh refinement to ensure a sharp interface between the fluids. To compute the interface dynamics the level set method (LSM) is used because of its ability to deal with merging and breakup. Also, the LSM is easy to extend to higher dimensions. Small cells arising from the cut-cell refinement are merged to improve the stability and performance. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
- …