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Open Access
January 1, 2001
### Abstract

In an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

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Open Access
January 1, 2001
### Abstract

Finite element approximations based on a penalty formulation of the elliptic obstacle problem are analyzed in the maximum norm. A posteriori error estimates, which involve a residual of the approximation and a spatially variable penalty parameter, are derived in the cases of both smooth and rough obstacles. An adaptive algorithm is suggested and implemented in one dimension.

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Open Access
January 1, 2001
### Abstract

A mathematical model of the linear sloshing in an infinite chute with a rectangular section filled by an inviscid incompressible liquid is considered. This model is an abstract second order differential equation in time with a singular integral operator coefficient on a free surface. In the case of a chute with a rib this operator coefficient is defined semi-explicitely through an auxiliary singular integral equation. Mathematical properties of the solution of the model are studied. The fundamental frequencies of the liquid as functions of the width and depth of the rib are investigated. The mathematical model is discretized by using the collocation quadrature method for the operator on the free surface and the Cayley transform method for the time derivative. This fully discrete model is investigated and the error estimates are obtained showing the spectral accuracy with respect to both the spatial and the time discretization parameters.

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Open Access
January 1, 2001
### Abstract

The paper deals with the stability, with respect to initial data, of difference schemes that approximate the heat-conduction equation with constant coefficients and nonlocal boundary conditions. Some difference schemes are considered for the one-dimensional heat-conduction equation, the energy norm is constructed, and the necessary and sufficient stability conditions in this norm are established for explicit and weighted difference schemes.

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Open Access
January 1, 2001
### Abstract

In this paper we investigate the stability of two-level operator-difference schemes in Hilbert spaces under perturbations of operators, the initial condition and right hand side of the equation. A priori estimates of the error are obtained in time- integral norms under some natural assumptions on the perturbations of the operators.

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Open Access
January 1, 2001
### Abstract

In the current paper the efficiency of the sparse-grid combination tech- nique applied to time-dependent advection-diffusion problems is investigated. For the time-integration we employ a third-order Rosenbrock scheme implemented with adap- tive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe±cient problem and a system of 2D non- linear Burgers' equations. In short, the combination technique proved more efficient than a single grid approach for the simpler linear problem. For the Burgers' equations this gain in efficiency was only observed if one of the two solution components was set to zero, which makes the problem more grid-aligned.

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Open Access
January 1, 2001
### Abstract

In this paper, a mortar finite element method is proposed for the Wilson nonconforming element. Multigrid method is used to solve the resulting discrete system.