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

Abstract

Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of the finite-element approach to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of function-value Dirichlet conditions prescribed on curvilinear boundaries method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard straight elements in the case of polygonal or polyhedral domains. However, besides obvious algebraic and geometric inconveniences, the isoparametric technique is helplessly limited in scope and simplicity, since its extension to degrees of freedom other than function values is not straightforward if not unknown. The purpose of this paper is to propose, study and test a simple alternative that bypasses all the above drawbacks, without eroding qualitative approximation properties. More specifically this technique can do without curved elements and is based only on polynomial algebra.

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

“…Now since ūh vanishes at three distinct points of γ e we can use a result in [22] according to which…”

“…In the case of complex non linear problems, this raises the delicate question on what numerical quadrature formula should be used to compute element matrices, in order to avoid qualitative losses in the error estimates or ill-posedness of approximate problems. In contrast, in the technique described in [25] and analyzed in [22] for two-dimensional problems, exact numerical integration can be used for the most common non linearities, since we only have to deal with polynomial integrands. Furthermore the element geometry remains the same as in the case of polytopic domains.…”

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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.

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“…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.…”