Computation of open Willmore-type surfaces Original Research Article

We present a numerical scheme for finding minimizing surfaces of functionals of the form View the MathML source where H and K are the mean and Gaussian curvatures, respectively, of the surface S, in the particular case of open surfaces represented as graphs. Such surfaces, known as Willmore-type surfaces, arise in various applications, such as biological membranes and shell mechanics. We construct a highly accurate finite element scheme using the “reduced quintic” finite element. As the problem is nonlinear, we use Newton iterations for subsequently solving the resulting nonlinear algebraic system. The initial guess (for starting the iterative process) is provided by solving a linearized version of the functional View the MathML source. We illustrate the performance of the scheme by numerical examples.