Optimal finite element methods of arbitrary order in curved domains approximated by the union of ordinary N-simplices

Vitoriano RUAS¹

¹ PUC-Rio, Brazil (CNPq research scholar)

doi:10.20906/CPS/CILAMCE2017-0210

Resumo

One of the reasons for the great success of the finite element method is its versatility to deal with different types of geometries. This is particularly true of problems posed in curved domains. In this case method's isoparametric version for higher order methods, in connection with meshes consisting of curved triangles or tetrahedra, has been mostly employed to recover the optimal approximation properties known to hold for ordinary elements, in the case of polytopic domains. However, besides geometric inconveniences, the isoparametric technique helplessly requires the manipulation of rational functions and the use of numerical integration to compute element matrices. In this work a simple alternative to bypass these drawbacks in the case of Dirichlet boundary conditions is presented. More precisely it is a technique based only on polynomial algebra, that can do without curved elements and does not erode qualitative approximation properties. Examples with classical Lagrange finite elements are shown for both two- and three-dimensional problems. An application with a Hermite finite element method also illustrates new method's universality. Moreover this technique can be combined with GFE or XFE methods to maintain their order in the case of a curved domain, whenever it is greater than one for a polygonal or a polyhedral domain.

Palavras-chave: curved domain; Dirichlet; finite elements; Hermite; interpolated boundary conditions; Lagrange; N-simplex