We study multigrid for solving the stochastic steady-state diffusion problem. We operate under the mild assumption that the diffusion coefficient takes the form of a finite Karhunen-Loéve expansion. The problem is discretized using a finite element methodology using the polynomial chaos method to discretize the stochastic part of the problem. We apply a multigrid algorithm to the stochastic pro...

A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fin...

A new discretization scheme for partial differential equations, based on the finite differences method, and its application to the two dimensional static diffusion equation is presented. This scheme produces better approximations than a standard use of finite differences. It satisfies properties of continuous differential operators and discrete versions of integral identities, which guarantee i...

We present a finite-element simulation tool for calculating light fields in 3D nano-optical devices. This allows to solve challenging problems on a standard personal computer. We present solutions to eigenvalue problems, like Bloch-type eigenvalues in photonic crystals and photonic crystal waveguides, and to scattering problems, like the transmission through finite photonic crystals. The discre...

Staggered grid finite volume methods (also called central schemes) were introduced in one dimension by Nessyahu and Tadmor in 1990 in order to avoid the necessity of having information on solutions of Riemann problems for the evaluation of numerical fluxes. We consider the general case in multidimensions and on general staggered grids which have to satisfy only an overlap assumption. We interpr...

A new numerical finite difference code has been developed to solve a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell. A kind of the overset (Chimera) grid named “YinYang grid” is used for the spatial discretization. The grid naturally avoids the pole problems which are inevitable in the latitude-longitude grids. The code is applied to...

The subject of this paper are grid refinements arising in finite element methods and their embedding into grids. We concentrate on embeddings with dilation 1 and construct for a class of grid refinements their embeddings with optimal load. For general cases we show that the problem to determine the minimal load is NP-complete.

We present a practical regular grid wave traveltime calculation that is based on Huygens wavefront expansion and grid traveltime mapping. Wavefront expansion is carried out by finite difference approximations to the equations that are equivalent to the Eikonal equation with a fixed time interval. Mapping traveltime from wavefront to regular grids is based on a dynamic ray tracing paraxial appro...

In this follow up paper to our previous study in [2], we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper [2] we ...

In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution ob...