Adaptation of EDP Solutions

The presentation is a result of my Doctoral Thesis in Mathematics. The article is developed in the area of Numerical Analysis, specifically Adaptativity and estimation of the error of solutions of partial differential equations. I hope that if you take a little time to read it, you may like and interest the subject.

Error estimation and adaptivity in mimic schemes for contour problems

(Estimación del error y adaptatividad en esquemas miméticos para problemas de contorno)

Abstract

In this article, we present the first adaptive process that defines an optimal mesh to calculate the solution of boundary problems using numerical mimetic methods; up to the present, there was no algorithm in the literature that fulfilled this purpose. The estimation of the error is made from the discrete version of the gradient operator. Numerical experimentation demonstrates the good behavior of the procedure when applied to contour problems.

Introduction

In the last two decades, a new type of finite difference conservative schemes, originally known as support operators, and later as mimetic methods, has shown advantages over classical finite difference schemes in solving various problems that arise in engineering. and the sciences. However, little or nothing has been said about how to define an adaptive process that yields a mesh that is optimal to find a solution that complies with a certain precision or error tolerance imposed by the problem to be solved.

A first attempt to define an adaptive process was given by Batista and Castillo in 2009. In this, a technique is proposed to construct the gradient and divergent operators on non-uniform structured meshes. However, to govern the adaptive process, a meshing function is defined, called fms(x) (mesh-size function), which is impractical, since to define this function, it is vital to know the regions where the largest will occur. errors or regions where more nodes should be placed. In other words, the function fms (·) is only useful to validate the good behavior of discrete operators in non-uniform structural meshes. Apart from this quote, the authors ignore the existence of any other reference with satisfactory results.

In this work, a new adaptive procedure is implemented, in the context of the mimetic operators developed by Castillo and Grone, which defines an optimal mesh to calculate the solution of stationary problems using non-uniform discrete operators. To define an estimate of the error made in the mimic approach, the mimetic discretization of the gradient operator is used without reference to the analytical solution of the boundary problem, which is generally unknown. In other words, an error estimate is made in the derivative of the solution, and not in the solution. It should be noted that this calculation does not merit any additional work, since the gradient has been previously defined to define the solution to the problem. Additionally, it can be said that the proposed process for estimating the error follows the ideas of the residual SPR estimator (superconvergent patch recovery) for the smoothing of tensions (derivatives), and in the case of displacements (problem unknown); both versions implemented for the case of the finite element method.

The error estimate and the adaptive algorithm are presented for one-dimensional problems. The extension to multidimensional static problems in structured meshes is in the process of development. On the other hand, the procedure for the case of unstructured meshes of quadrilateral elements represents, nowadays, an open problem.