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In recent papers the author introduced a simple alternative to isoparametricfinite elements of the n-simplex type, to enhance the accuracy ofapproximations of second-order boundary value problems with Dirichletconditions, posed in smooth curved domains. This technique is based upontrial-functions consisting of piecewise polynomials defined on straight-edgedtriangular or tetrahedral meshes, interpolating the Dirichlet boundaryconditions at points of the true boundary. In contrast the test-functions aredefined upon the standard degrees of freedom associated with the underlyingmethod for polytopic domains. While method's mathematical analysis for bothsecond- and fourth-order problems in two-dimensional domains was carried out inarxiv NA-1701.00663 and in a submitted paper, this article is devoted to thestudy of the three-dimensional case, in which the method is nonconforming.Well-posedness, uniform stability and optimal a priori error estimates in theenergy norm are demonstrated for a tetrahedron-based Lagrange family of finiteelements. Novel L2-error estimates for the class of problems considered in thiswork are also proved. A series of numerical examples illustrates the potentialof the new technique. In particular its better accuracy at equivalent cost ascompared to the isoparametric technique is highlighted. Moreover the greatgenerality of the new approach is exemplified through a method with degrees offreedom other than nodal values.

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