Jose L. Gracia

A parameter robust numerical method for a two dimensional reaction-diffusion problem

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation

A reaction-diffusion problem with a Caputo time derivative of order

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

\alpha\in (0,1)

Fourier Analysis for Multigrid Methods on Triangular Grids

is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time

High order methods for elliptic and time dependent reaction-diffusion singularly perturbed problems

t=0

A finite difference method for a two-point boundary value problem with a Caputo fractional derivative

, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

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.

A coupled system of singularly perturbed parabolic reaction-diffusion equations

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 defect–correction parameter-uniform numerical method for a singularly perturbed convection–diffusion problem in one dimension

The objective of this paper is to construct some high order uniform numerical methods to solve linear reaction-diffusion singularly perturbed problems. First, for 1D elliptic problems, based on the central finite difference scheme, a new HODIE method is defined on a piecewise uniform Shishkin mesh. Using this HODIE scheme jointly with a two stage SDIRK method, we solve a 1D parabolic singularly perturbed problem. In both cases we prove that the methods are third-order uniform convergent in the maximum norm. Finally, for a 2D parabolic problem of the same type, we show numerically that the combination of the HODIE scheme with a fractional step RK method gives again a third-order uniform convergent scheme.

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

A two-point boundary value problem whose highest order term is a Caputo fractional derivative of order δ∈(1, 2) is considered. Al-Refais comparison principle is improved and modified to fit our problem. Sharp a priori bounds on derivatives of the solution u of the boundary value problem are established, showing that u″(x) may be unbounded at the interval endpoint x=0. These bounds and a discrete comparison principle are used to prove pointwise convergence of a finite difference method for the problem, where the convective term is discretized using simple upwinding to yield stability on coarse meshes for all values of δ. Numerical results are presented to illustrate the performance of the method.