New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and confo...

It is announced that the Maxwell equations can be solved on unstructured grids using finite integration methods. Numerical experiments show that next to the finite-volume method, a discretization technique can be defined based on ’finite-surface’ integration.

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let Gc denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement Gc > 2. However, in examples the requirement i...

ii Preface Radial Basis Function (RBF) methods have become the primary tool for interpolating multidimensional scattered data. RBF methods also have become important tools for solving Partial Differential Equations (PDEs) in complexly shaped domains. Classical methods for the numerical solution of PDEs (finite difference, finite element, finite volume, and pseudospectral methods) are based on p...

The signed distance function d to an embedded (hyper-) surface Γ is required in the analysis and implementation of some higher order methods for the numerical treatment of partial differential equations on surfaces. Two algorithms for the approximation of d are presented in this paper, which only require a finite element approximation of a (smooth) level set function of Γ. One method is based o...

Two-level finite element methods are applied to solve numerically the 2D/3D steady Navier-Stokes equations if a strong uniqueness condition ( ‖f‖ −1 ‖f‖0 ) 1 2 ≤ δ = 1 − N‖f‖ −1 ν2 holds, where N is defined in (2.4)-(2.6). Moreover, one-level finite element method is applied to solve numerically the 2D/3D steady Navier-Stokes equations if a weak uniqueness condition 0 < δ < ( ‖f‖ −1 ‖f‖0 ) 1 2 ...