Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

Abstract

In recent papers (see e.g. [22], [25], [26]) the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of secondorder boundary value problems with Dirichlet conditions, posed in smooth curved domains. This technique is based upon trial-functions consisting of piecewise polynomials defined on straightedged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast the test-functions are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. While method's mathematical analysis for both second-and fourth-order problems in two-dimensional domains was carried out in [22] and [27], this paper is devoted to the study of the three-dimensional case, in which the method is nonconforming. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are demonstrated for a tetrahedron-based Lagrange family of finite elements. Unprecedented L 2 -error estimates for the class of problems considered in this work are also proved. A series of numerical examples illustrates the potential of the new technique. In particular its better accuracy at equivalent cost as compared to the isoparametric technique is highlighted. Moreover the great generality of the new approach is exemplified through a method with degrees of freedom other than nodal values.

Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique

“…We conclude in Section 5 with some comments on the methodology studied in this work. In particular we briefly show that the technique addressed in Sections 2 and 3 applies with no particular difficulty to the case of boundary value problems posed in curved three-dimensional domains (see also [17]).…”

“…Nonetheless their proof is at the price of several additional technicalities, especially in the non convex case. We address all those issues more thoroughly in [17].…”