  In recent papers the author introduced a simple alternative to isoparametric
finite elements of the n-simplex type, to enhance the accuracy of
approximations of second-order boundary value problems with Dirichlet
conditions, posed in smooth curved domains. This technique is based upon
trial-functions consisting of piecewise polynomials defined on straight-edged
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
arxiv NA-1701.00663 and in a submitted paper, this article 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. Novel L2-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.
