Numerical algorithms for the biharmonic equation over smooth domains

Using the Goursat representation for the biharmonic function and approximate solutions of a corrected Muskhelishvili equation we construct approximate solutions for biharmonic problems in smooth domains of $\mathbf{R}^2$. We make use of suitable spline spaces to develop the Galerkin, collocation and qualocation methods. The sequence of the approximate solutions constructed with these methods is shown to converge uniformly on each compact subset of the original domain $D$. Under additional conditions it converges uniformly on $D$. Our extensive numerical experience confirms the good stability behavior of all the proposed methods. Convergence is achieved also within the framework of the theoretical findings. Several examples illustrate the versatility of the algorithms for various contours, including those with high curvature.