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Open Access
January 1, 2001
### Abstract

In this paper we show how the theory of interpolation of function spaces can be used to establish convergence rate estimates for finite difference schemes on nonuniform meshes. As a model problem we consider the first boundary value problem for the Poisson equation. Using the interpolation theory we construct a fractional-order convergence rate estimate which is consistent with the smoothness of data.

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Open Access
January 1, 2001
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In this paper we describe methods to approximate functions and dif- ferential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the pos- sibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.

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Open Access
January 1, 2001
### Abstract

The existence of a solution of the finite-dimensional problem with continuous M-mappings and multivalued diagonal maximal monotone operators on an ordered interval, which is formed by the so-called subsolution, is proved. Under several additional assumptions on the operators the monotone dependence of a solution upon the right-hand side is investigated. This result implies, in particular, the uniqueness of a solution and serves as a basis for the analysis of the convergence for a multisplitting iterative method. As an illustrative example, the finite difference scheme approximating a model variational inequality is studied by using the general results.

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Open Access
January 1, 2001
### Abstract

In this work, a stability of three-level operator-difference schemes on nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long time stability (for t → ∞) in the sense of the initial data and the right-hand side are obtained in different energy norms without demanding the quasiuniformity of the grid. New difference schemes of the second order of local approximation on nonuniform grids both in time and space on standard stencils for parabolic and wave equations are adduced.

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Open Access
January 1, 2001
### Abstract

This paper is devoted to the studies of the properties of the solutions of non-linear partial differential equation, being basic ones in differential formulation of the magnetostatic problem of finding the magnetic field dis- tribution. The question of the existence of solutions, possessing an unlimited gradient, for this equation is of particular interest. Previous works dealt with the linear equation type, and also a boundary value problem was con- sidered for certain requirements for the µ function, as well as a more general non-linear case was studied. It was shown that such solutions exist, and their properties will be investigated. The difference scheme for the boundary value problem was built in the domain with corner and numerical calculations were given.

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Open Access
January 1, 2001
### Abstract

In this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.