Preprocessing for Finite Element Discretizations of Geometric Problems

In this paper, we use finite element method to approximate the solutions of parameter-dependent geometric problems, and investigate the possibility of using symbolic methods as a preprocessing step. The main idea of our approach is to construct suitable finite element discretizations of the nonlinear elliptic equations leading to systems of algebraic equations, which can be subsequently solved by symbolic computation within the tolerance of computer algebra software. The prolongation of the preprocessed symbolic solution can serve as a starting value for a numerical iterative method on a finer grid. A motivation for this approach is that usual numerical iterations (e.g. via Newton-type or fixed-point iterations) may diverge if no appropriate initial values are available. Moreover, such a purely numerical approach will not find all solutions of the discretized problem if there are more than one. A final motivation for the use of symbolic methods is the fact that all discrete solutions can be obtained as functions of unknown parameters. In this paper, we focus on a special class of partial differential equations derived from geometric problems. A main challenge in this class is the fact that the polynomial structure of the nonlinearity is not explicit in the divergence form usually used for finite element discretization. As a consequence, the discrete form would always yield some non-polynomial terms. We therefore consider two different discretizations, namely a polynomial reformulation before discretization and a direct discretization of the divergence form with polynomial approximation of the discrete system. In order to perform a detailed analysis and convergence theory of the discretization methods we investigate some model problems related to mean-curvature type equations.