Francisco Javier Lisbona

A finite difference analysis of Biot's consolidation model

In this paper, stability estimates and convergence analysis of finite difference methods for the Biots consolidation model are presented. Initially central differences for space discretization and a weighed two-level time scheme are analyzed. To improve some stability and convergence limitations for this scheme we also consider space discretizations on MAC type grids (staggered grids). Numerical results are given to illustrate the obtained theoretical results.

A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems

In this paper we construct a numerical method to solve one-dimensional time-dependent convection-diffusion problem with dominating convection term. We use the classical Euler implicit method for the time discretization and the simple upwind scheme on a special nonuniform mesh for the spatial discretization. We show that the resulting method is uniformly convergent with respect to the diffusion parameter. The main lines for the analysis of the uniform convergence carried out here can be used for the study of more general singular perturbation problems and also for more complicated numerical schemes. The numerical results show that, in practice, some of the theoretical compatibility conditions seem not necessary.

A fractional step method on a special mesh for the resolution multidimensional evolutionary convection-diffusion problems

In this paper we consider numerical schemes for multidimensional evolutionary convection-diffusion problems, where the approximation properties are uniform in the diffusion parameter. In order to obtain an efficient method, to provide good approximations with independence of the size of the diffusion parameter, we have developed a numerical method which combines a finite difference spatial discretization on a special mesh and a fractional step method for the time variable. The special mesh allows a correct approximation of the solution in the boundary layers, while the fractional steps permits a low computational cost algorithm. Some numerical examples confirming the expected behavior of the method are shown.

High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type

In this paper we construct and analyze two compact monotone finite difference methods to solve singularly perturbed problems of convection–diffusion type. They are defined as HODIE methods of order two and three, i.e., the coefficients are determined by imposing that the local error be null on a polynomial space. For arbitrary meshes, these methods are not adequate for singularly perturbed problems, but using a Shishkin mesh we can prove that the methods are uniformly convergent of order two and three except for a logarithmic factor. Numerical examples support the theoretical results.

Fourier Analysis for Multigrid Methods on Triangular Grids

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.

A coupled system of singularly perturbed parabolic reaction-diffusion equations

In this paper systems with an arbitrary number of singularly perturbed parabolic reaction-diffusion equations are examined. A numerical method is constructed for these systems which involves an appropriate layer-adapted piecewise-uniform mesh. The numerical approximations generated from this method are shown to be uniformly convergent with respect to the singular perturbation parameters. Numerical experiments supporting the theoretical results are given.

An Efficient Multigrid Solver based on Distributive Smoothing for Poroelasticity Equations

In this paper, we present a robust distributive smoother in a multigrid method for the system of poroelasticity equations. Within the distributive framework, we deal with a decoupled system, that can be smoothed with basic iterative methods like an equation-wise red-black Jacobi point relaxation. The properties of the distributive relaxation are optimized with the help of Fourier smoothing analysis. A highly efficient multigrid method results, as is confirmed by Fourier two-grid analysis and numerical experiments.

High Order varepsilon-Uniform Methods for Singularly Perturbed Reaction-Diffusion Problems

The central difference scheme for reaction-diffusion problems, when fitted Shishkin type meshes are used, gives uniformly convergent methods of almost second order. In this work, we construct HOC (High Order Compact) compact monotone finite difference schemes, defined on a priori Shishkin meshes, uniformly convergent with respect the diffusion parameter ?, which have order three and four except for a logarithmic factor. We show some numerical experiments which support the theoretical results.

Contractivity results for alternating direction schemes in Hilbert spaces

Abstract In this paper we obtain some contractivity results for operators R (− A 1 ,…,− A n ), where R ( z 1 ,…, z n ) are rational approximations to exp( z 1 +⋯+ z n ), and A i are maximal monotone operators on a Hilbert space H . A general result is proved by using an extension to several variables of a result of Von Neumann for bounding f ( A )( f a holomorphic function, A an operator on H ). This theory is applied to the convergence analysis of Alternating Direction methods and, more generally, to Fractional Steps schemes.

Some numerical experiments with multigrid methods on Shishkin meshes

Piecewise uniform meshes introduced by Shishkin, are a very useful tool to construct robust and efficient numerical methods to approximate the solution of singularly perturbed problems. For small values of the diffusion coefficient, the step size ratios, in this kind of grids, can be very large. In this case, standard multigrid methods are not convergent. To avoid this troublesome, in this paper we propose a modified multigrid algorithm, which works fine on Shishkin meshes. We show some numerical experiments confirming that the proposed multigrid method is convergent, and it has similar properties that standard multigrid for classical elliptic problems.