Solving Laplace-Beltrami Equation on Parametric Surface by Rational Bézier Elements

When solving the partial differential equations on manifold surfaces by the finite element method, the accuracy of the numerical results may be seriously reduced by the geometrical errors which are caused by the approximation of the computational domains with the traditional polygonal elements.The geometric accurate finite element method is presented to remedy the issues by the parameterization of the geometrical domains with the rational Bernstein polynomials.Firstly, the new knots are repeatedly inserted into the parametric interval to convert the NURBS surface to the rational Bézier elements without changing it geometrically or parametrically.Then, the Galerkin method is employed to establish the equivalent weak forms for the second-order elliptic partial differential equations which involve the Laplace-Beltrami operator on the parametric surface.It's a difficult task to enforce the Dirichlet boundary conditions in the presented method because of the non-interpolation properties of the Bernstein functions.Therefore, the collocation method is employed to solve it, and the optimal convergence is acquired.Finally, the numerical examples show that the method can effectively reduce the discretization errors and improve the accuracy of the numerical results.