Non-matching Grids Research Articles

Efficient iterative solvers for elliptic finite element problems on nonmatching grids

Efficient iterative solvers for elliptic finite element problems on nonmatching grids

Domain decomposition with nonmatching grids: augmented Lagrangian approach

We propose and study a domain decomposition method which treats the constraint of displacement continuity at the interfaces by augmented Lagrangian techniques and solves the resulting problem by a parallel version of the Peaceman-Rachford algorithm. We prove that this algorithm is equivalent to the fictitious overlapping method introduced by P.L. Lions. We also prove its linear convergence independently of the discretization step h, even if the finite element grids do not match at the interfaces. A new preconditioner using fictitious overlapping and well adapted to three-dimensional elasticity problems is also introduced and is validated on several numerical examples.

Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure

Maxwell's equations in the time domain are first cast into a conservation form. They are then solved using a finite-volume discretization procedure proven very successful in solving some hyperbolic partial differential equations in computational fluid dynamics such as Euler or Navier-Stokes equations. The Lax-Wendroff explicit scheme is used to solve the discrete Maxwell's equations and as a result, second-order accuracy is achieved in both time and space. Multizoning is used to facilitate treatment of objects with internal and/or external sophistication. Body-fitted coordinates are used to map each zone independently from the rest of the object, allowing for nonmatching grid lines at the interface. Body-fitted coordinates facilitate accurate implementation of various boundary conditions on the object. The density of the computational mesh can vary and may be specified according to the local wavelength in the material. Also in the exterior region and away from the body, the mesh density is gradually decreased to reduce the total number of the grid points and the unknown field vectors. The formulation accounts for any variations in the material properties as a function of space, time, or frequency. In addition to perfect conductors, impedence surfaces and resistive sheets are handled. At the outer boundary of the computational domain a first-order, nonreflecting outer boundary condition is used. The Lax-Wendroff scheme, while providing second-order accuracy in the time and space, does not require, for its stability, an interlaced mesh for the electric and magnetic fields or the material properties. The method can be used for both time-harmonic and pulse excitations. At the end of the time-marching, the computed field in the time domain near the object and tangential to a conveniently chosen contour is stored. This information is converted into the frequency domain using fast Fourier transforms. If far field quantities such as radar cross-section or antenna radiation pattern are desired, they may be computed from the frequency-domain information using a standard free-space Green's function.

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation