Volume 16, Issue 3

We use sampling techniques to compute eigenvalues of discontinuous second order boundary value problems. The eigenvalues are in general complex numbers and are not necessarily simple. Therefore error estimates for the truncation and amplitude errors on ℂ of the sampling expansion and its termwise derivative are used. Moreover the problem we consider has two parts throughout [–1,1] and is not in general self adjoint. The boundary and compatibility conditions are assumed to be regular in the sense of Birkhoff to guarantee the existence of eigenvalues.

The main goal of this article is two fold: (i) To discuss a methodology for the numerical solution of the Dirichlet problem for a Pucci equation in dimension two. (ii) Use the ensuing algorithms to investigate the homogenization properties of the solutions when a coefficient in the Pucci equation oscillates periodically or randomly in space. The solution methodology relies on the combination of a least-squares formulation of the Pucci equation in an appropriate Hilbert space with operator-splitting techniques and mixed finite element approximations. The results of numerical experiments suggest second order accuracy when globally continuous piecewise affine space approximations are used; they also show that the solution of the problem under consideration can be reduced to a sequence of discrete Poisson–Dirichlet problems coupled with one-dimensional optimization problems (one per grid point).

In this paper, we consider the one-dimensional modified Burgers' equation in the finite domain. This type of problem arises in the field of sonic boom and explosions theory. At the high Reynolds' number there is a boundary layer in the right side of the domain. From the numerical point of view, one of the difficulties in dealing with this problem is that even smooth initial data can give rise to solution varying regions, i.e., boundary layer regions. To tackle this situation, we propose a numerical method on non-uniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed numerical method comprises of Euler implicit and upwind finite difference scheme. First we discretize in the temporal direction by means of Euler implicit method which yields the set of ordinary differential equations at each time level. The resulting set of differential equations are approximated by upwind scheme on Shishkin mesh. The proposed method has been shown to be parameter uniform and of almost first order accurate in the space and time. An extensive amount of analysis has been carried out in order to prove parameter uniform convergence of the method. some test examples have been solved to verify the theoretical results.

In this paper, a cascadic multigrid method for P 1 -nonconforming quadrilateral finite element approximation of second order elliptic problem is studied. A new intergrid transfer operator is constructed to connect the nonnested coarse and fine grid spaces. We show that the cascadic multigrid method is accurate with optimal complexity for conjugate gradient smoother, and suboptimal complexity for some other traditional multigrid smoothers. Finally, some numerical experiments are reported to support the theory.