PGD for Solving the Biharmonic Equation

Abstract

Biharmonic problem has been raised in many research fields, such as elasticity problem in plate geometries or the Stokes flow problem formulated by using the stream function. The fourth order partial differential equation can be solved by applying many techniques. When using finite elements C 1 continuity must be assured. For this purpose Hermite interpolations constitute an appealing choice, but it imply the consideration of many degrees of freedom at each node with the consequent impact on the resulting discrete linear problem. Spectral approaches allow exponential convergence whilst a single degree of freedom is needed. However, the enforcement of boundary conditions remains a tricky task. In this paper we propose a separated representation of the stream function which transform the 2D solution in a sequence of 1D problems, each one be solved by using a spectral approximation. INTRODUCTIONThe The spectral method has been widely used in the solution of Partial Differential Equations -PDE -, in particular high order PDEs, e.g. [7].The Chebyshev spectral collocation method [8,9] has been traditionally used to solved biharmonic problems. Its main advantage lies in the fact that it only needs a degree of freedom per node and it exhibits exponential convergence rates.In this paper spectral collocation schemes are combined with the PGD technique [10,11] that allows a separated representation of the fields involved in the model, and then in our case, transform the solution of a 2D model into the solution of few 1D problems.