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SUMMARYIn this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one-dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique.

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In this paper a local Fourier analysis technique for multigrid methods on triangular grids is presented. The analysis is based on an expression of the Fourier transform in new coordinate systems, both in space variables and in frequency variables, associated with reciprocal bases. This tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids. Different smoothers for the discrete Laplace operator obtained with linear finite elements are analyzed. A new three-color smoother has been studied and has proven to be the best choice for "near" equilateral triangles. It is also shown that the block-line smoothers are more appropriate for irregular triangles. Numerical test calculations validate the theoretical predictions.

In this paper variable-stepsize, variable-formula, multistep methods for the numerical solution of ordinary differential equations are considered. Some more general results on stability than are given in the papers ] are obtained. Further, the asymptotic behavior of the error in the variable-stepsize, fixed-formula case is studied. Finally, the above study is extended to predictorcorrector methods.