Rakhim Aitbayev (New Mexico Tech)

ABSTRACT:

The finite element method (FE) is one of the most widely used numerical methods for solving partial differential equations. The method approximates the solution of a boundary value problem (BVP) by a piecewise-polynomial function, which is a solution of a finite dimensional analogue of the so-called variational form of the BVP. To form a corresponding linear system, one needs to evaluate a large number of integrals. Replacing these integrals by numerical quadratures, gives the so-called quadrature finite element Galerkin method. In this presentation, we approximate a Dirichlet BVP with a biharmonic equation with a quadrature Galerkin scheme based on the two-point Gaussian quadrature. We will present convergence results using two techniques: the standard approach based on the first Strang lemma and the one based on the equivalent orthogonal spline collocation formulation.