B-splines Bases Functions for the Finite Element Approximations

Abstract

In this thesis we discuss the necessary background material for the description of spline-based finite element methods. We explain the basic finite element idea by constructing the classical Ritz-Galerkin scheme for Poisson’s equation. We define the concept of ellipticity and the Lax-Milgram existence theorem for variational problems of the Poisson’s equation. We introduce and define the concept of splines and B-splines. We construct the fundamental recurrence relation, which allows us to evaluate B-splines efficiently and to compute their polynomial segments. We alsodiscuss algorithms for grid refinement and computation of scalar products for B-splines and their derivative. Then we construct the finite element bases functions in regular grids using B-splines and multivariate B-splines. Finally, we discuss the approximation of Poisson’s equation with essential and natural boundary conditions.