Polynomial basis functions for curved elements using hyperbolas

AbstractA conforming polynomial second order basis for the three sided two-dimensional finite elements with one curved side is constructed in such a way that the curved side is approximated by an arc of hyperbola. The basis is used to calculate approximate solutions of Laplace's equation over the unitary disk with Dirichlet boundary conditions. The basis has the property that it remains conforming when the curved side reverts to a straight line segment. The calculations of the typical integrals are made directly in the original domain of interest without the use of a non-linear transformation that is required in the high order transformation methods. Various tesselations of the problem domain were done and the numerical experiments show that the results are completely satisfactory for all the examples considered