Solución de ecuaciones diferenciales elípticas en regiones planas irregulares usando mallas convexas generadas por métodos variacionales empleando elementos finitos

Latest revision as of 10:58, 14 June 2017

Abstract

Recently, in order to approximate the solution of a partial differential equation overa n irregular planar domains, several efficient and robust variational methods designed to generate smooth and convex grids on such regions have been proposed (3-9,11-14). For those grids, several schemes have also been designed, and for them it is quite clear how effortless the use of the grid logical rectangular data structure can be (1,2,15). This fact makes these schemes attractive competitors to the finite element methods, which use unstructured grids and, in consequence, non trivial data structures inorder to save the grid information. Nevertheless, one must acknowledge that, since triangulatiolns have been known for a while , finite element methods have been known for a while, finite element methods have been thoroughly studies, and there is a lot of research on how to assemble the systems required to solve a large class of equations. Thus, a question that arises in a natural way is how competitive are FE/FD methods, when applied to the structured convex grids generated for irregular regions-which often have elongated elements-, in order to produce the numerical solution in an easy computational way using structured grids and, at the same time, accurate enough by using finite elements. In this paper we show how to accomplish this goal, and a series of numerical examples at the end provided a good example of the validity of the approach.

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 == Abstract ==
-Recientemente, con el objeto de ser usadas para aproximar la solución de ecuaciones diferenciales parciales en dominios de forma irregular empleando diferencias finitas, se han propuesto varios métodos variacionales eficientes y robustos para generar mallas estructuradas, convexas y suaves que funcionan bien en dichas regiones (3-9,11-14). Para esas mallas, se han desarrollado también algunos esquemas de los cuales destaca la facilidad computacional que implica el usar uns estructura lógicamente rectangular (1,2,15). Este hecho los convierte en una alternativa de interés a los métodos de elementos finitos que emplean mallas no estucturadas, pues estas últimas tienen el inconveniente de que su programación requiere con frecuencia de una estructura de datos compleja. Sin embargo, hay que reconocer que, dado que la triangulación de Delaunay se conoce de tiempo atrás, los métodos de elemento finito tienen la ventaja de que se ha estudiado el problema en muchos contextos y existe abundante literatura que describe como ensamblar eficientemente sistemas para aproximar la solución de una gran variedad de ecuaciones. Así surge de manera natural la pregunta de qué tan competitivos son los elementos y/o diferencias finitos en las mallas estructuradas generadas variacionalmente en regiones muy irregulares-y que con frecuencia tienen elementos elongados para obtener una solución numérica en forma computacionalemente sencilla empleando mallas estructuradas y al mismo tiempo con precisión razonable empleando elementos finitos. En este trabajo mostramos como lograr este objetivo, y una serie de experimentos numéricos empleando mallas en regiones muy irregulares muestran la eficiencia del enfoque propuesto. Summary Recently, in order to approximate the solution of a partial differential equation overa n irregular planar domains, several efficient and robust variational methods designed to generate smooth and convex grids on such regions have been proposed (3-9,11-14). For those grids, several schemes have also been designed, and for them it is quite clear how effortless the use of the grid logical rectangular data structure can be (1,2,15). This fact makes these schemes attractive competitors to the finite element methods, which use unstructured grids and, in consequence, non trivial data structures inorder to save the grid information. Nevertheless, one must acknowledge that, since triangulatiolns have been known for a while , finite element methods have been known for a while, finite element methods have been thoroughly studies, and there is a lot of research on how to assemble the systems required to solve a large class of equations. Thus, a question that arises in a natural way is how competitive are FE/FD methods, when applied to the structured convex grids generated for irregular regions-which often have elongated elements-, in order to produce the numerical solution in an easy computational way using structured grids and, at the same time, accurate enough by using finite elements. In this paper we show how to accomplish this goal, and a series of numerical examples at the end provided a good example of the validity of the approach.
+Recently, in order to approximate the solution of a partial differential equation overa n irregular planar domains, several efficient and robust variational methods designed to generate smooth and convex grids on such regions have been proposed (3-9,11-14). For those grids, several schemes have also been designed, and for them it is quite clear how effortless the use of the grid logical rectangular data structure can be (1,2,15). This fact makes these schemes attractive competitors to the finite element methods, which use unstructured grids and, in consequence, non trivial data structures inorder to save the grid information. Nevertheless, one must acknowledge that, since triangulatiolns have been known for a while , finite element methods have been known for a while, finite element methods have been thoroughly studies, and there is a lot of research on how to assemble the systems required to solve a large class of equations. Thus, a question that arises in a natural way is how competitive are FE/FD methods, when applied to the structured convex grids generated for irregular regions-which often have elongated elements-, in order to produce the numerical solution in an easy computational way using structured grids and, at the same time, accurate enough by using finite elements. In this paper we show how to accomplish this goal, and a series of numerical examples at the end provided a good example of the validity of the approach.
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