  One of the reasons for the 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 of arbitrary shape. In the case of
second order boundary-value problems with 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 methods of order greater than one
based on standard straight-edged elements, in the case of polygonal or
polyhedral domains. However, besides algebraic and geometric inconveniences,
the isoparametric technique is limited in scope, since its extension to degrees
of freedom other than function values is not straightforward. The purpose of
this paper is to study a simple alternative that bypasses the above drawbacks,
without eroding qualitative approximation properties. Among other advantages,
this technique can do without curved elements and is based only on polynomial
algebra. It is first illustrated in the case of the convection-diffusion
equation solved with standard Lagrange elements. Then it is applied to the
solution with Hermite elements of the biharmonic equation with Dirichlet
boundary conditions.
