Francisco José Gaspar

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.

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.

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.

A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

We consider the multigrid solution of the generalized Stokes equations with a segregated (i.e., equationwise) Gauss--Seidel smoother based on a Uzawa-type iteration. We analyze the smoother in the framework of local Fourier analysis, and obtain an analytic bound on the smoothing factor showing uniform performance for a family of Stokes problems. These results are confirmed by the numerical computation of the two-grid convergence factor for different types of grids and discretizations. Numerical results also show that the actual convergence of the W-cycle is approximately the same as that obtained by a Vanka smoother, despite this latter smoother being significantly more costly per iteration step.

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.

Distributive smoothers in multigrid for problems with dominating grad–div operators

In this paper, we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating grad–div operator. In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the mesh sizes in space and time. The applications range from model problems to secondary consolidation Biots model. We focus on the smoothing issue and mainly solve academic problems on Cartesian-staggered grids. Copyright

Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis

In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented. Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed. A four-color smoother is presented as an efficient choice for regular tetrahedral grids, whereas line and plane relaxations are needed for poorly shaped tetrahedra. A novel partitioning of the Fourier space is proposed to analyze the four-color smoother. Numerical test calculations validate the theoretical predictions. A multigrid method is constructed in a block-wise form, by using different smoothers and different numbers of pre- and post-smoothing steps in each tetrahedron of the coarsest grid of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.

Accuracy Measures and Fourier Analysis for the Full Multigrid Algorithm

The full multigrid (FMG) algorithm is often claimed to achieve so-called discretization-level accuracy. In this paper, this notion is formalized by defining a worst-case relative accuracy measure, denoted

Multigrid finite element methods on semi‐structured triangular grids for planar elasticity

E_{FMG}^{\ell}

Multigrid fourier analysis on semi‐structured anisotropic meshes for vector problems

, which compares the total error of the