Treatment of curved boundary value problem using finite element method

Abstract

In this paper, the curved boundary value problem is discussed. We consider the problem of flow past a cylinder, since the analytic solution exists and also the symmetry of the problem allows a detailed numerical treatment of a small portion of the flow fluid. The governing equation is Laplace's equation for either the stream function or the potential function, each has its boundary conditions. A smooth mesh variation is used with linear, quadratic and Hermitian cubic elements. Errors and error per cent are calculated on and away from the cylinder at the same point position as in the linear, quadratic and cubic elements. The results are satisfactory. The problem of curved boundaries is treated again by introducing elements which have one or more curved sides. The curved elements are handled by employing isoparametric transformation. The isoparametric coordinates which carry out the required transformation are obtained through point transformation. The results are satisfactory and the errors are extremely small comparing with both the linear and Hermitian elements with fine mesh for the polynomial region. This indicates the importance of boundary modelling.