Abstract.We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.

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We use higher dimensional B-splines as basis functions to find the approximations for the Dirichlet problem of the Poisson equation in dimension two and three. We utilize the boundary data to remove unnecessary bases. Our method is applicable to more general linear partial differential equations. We provide new basis functions which do not require as many B-splines. The number of new bases coincides with that of the necessary knots. The reducing process uses the boundary conditions to redefine a basis without extra artificial assumptions on knots which are outside the domain. Therefore, more accuracy would be expected from our method. The approximation solutions satisfy the Poisson equation at each mesh point and are solved explicitly using tensor product of matrices.

Abstract.In this paper we investigate the behavior of the solution ofSolutions of this equation are considered to be approximations to the solutions of the corresponding parabolic conservation laws. We obtain decay results on the norms of the dilference between the solution for L infinite and the solution when L is finite.