This article is two fold. Firstly, we derive optimal order a priori error estimates for Babuška's penalty method applied to inhomogeneous Dirichlet problem. Secondly, we derive convergence of W-cycle and V-cycle multigrid algorithms for the resulting system. To this end, a simple pre-conditioner is introduced to remedy the ill-condition due to over-penalty.

We study the BV-α-Deconvolution model. It is a family of regularizations of the Barotropic Vorticity (BV) model that generalize the BV-α model and improve its accuracy. A both unconditionally stable and optimally convergent scheme for the BV-α-Deconvolution model is proposed and we show that it is O(α^{2N+2}), where N is the deconvolution order, whereas the BV-α model is at most second order accurate. We perform numerical simulations to confirm the predicted convergence rates and test the model in the traditional double gyre wind experiment. For the latter test, we show that the BV-α-Deconvolution model can retrieve the expected high resolution pattern being more accurate for larger values of deconvolution order.

In this paper, we present a study of some high-order compact difference schemes for solving the Fitzhugh-Nagumo equations governed by two coupled time-dependent nonlinear reaction diffusion equations in two variables. Solving the Fitzhugh-Nagumo equations is quite challenging, since the equations involve spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed have sixth order accuracy in space, and fourth order in time if the fourth order Runge-Kutta method is adopted for time marching. To improve effciency, we also propose an ADI scheme (for two dimensional problems), which has second order accuracy in time. Numerical results are presented for plane wave propagation in one dimension and spiral waves for two dimensions.

The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on the Shishkin mesh is analyzed. A new, conceptually simple proof of pointwise convergence uniform in the perturbation parameter is provided. The proof is based on the preconditioning of the discrete system.

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential nonconvergence and instability.

This paper presents the multiscale analysis and numerical algorithms for the convectiondiffusion equations with rapidly oscillating periodic discontinuous coefficients. The multiscale asymptotic expansions are developed and an explicit rate of convergence is derived for the convex domains. An efficient multiscale hybrid FEM-FVM algorithm is constructed, and numerical experiments are reported to validate the predicted convergence results.

We consider the coupled system modeling the interaction of time-harmonic progres- sive water-waves with an array of three-dimensional freely oating obstacles in a two-layer uid. Presenting a variational and operator formulation for the problem and a condition guaranteeing the existence of trapped waves, we give a number of examples of oating structures supporting trapped waves. We also study how the problem parameters (density ratio, obstacle dimensions, layer depths and radian frequency) in uence the trapping condition.