In recent papers (see e.g. [17] and [18]) a nonparametric technique of the Petrov-Galerkin type was analyzed, whose aim is the accuracy enhancement of higher order finite element methods to solve boundary value problems with Dirichlet conditions, posed in smooth curved domains. In contrast to parametric elements, it employs straight-edged triangular or tetrahedral meshes fitting the domain. In order to attain best-possible orders greater than one, piecewise polynomial trial-functions are employed, which interpolate the Dirichlet conditions at points of the true boundary. The test-functions in turn are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. As a consequence, when the problem at hand is self-adjoint a non symmetric linear system has to be solved. This paper is primarily aimed at showing that in this case, an efficient symmetrization of the solution procedure can be achieved by means of a fast converging iterative method. In order to illustrate the great generality of our nonparametric approach,experimentation is presented with a finite element method having degrees of freedom other than nodal values. More specifically we consider a nonconforming quadratic element in thesolution of the three-dimensional Poisson equation. The performance evaluation however is conducted as well for two versions of the classical conforming quadratic method, namely, the nonparametric Petrov-Galerkin formulation considered in [18] and the standard isoparametric one. The study of this symmetrization is completed by an optimal error estimationin the broken H1-norm for the nonparametric version of the nonconforming method, which had not been addressed in previous work.