Abstract
This chapter describes finite element approximation with uniform B-Splines. These techniques do not require any grid generation and yield highly accurate approximations with relatively few parameters. They are particularly well-suited in combination with spline-based geometric modeling systems. The finite element method provides a broad range of applications in continuum mechanics, fluid dynamics, field theory, and other areas in engineering and mathematical physics. The chapter describes the way to solve variational problems with finite dimensional approximations. The basic principle of finite element methods can be illustrated for the Poisson's equation on a bounded domain with homogeneous Dirichlet boundary condition. This chapter explains meshless methods that use weighted finite element subspaces. It discusses that uniform splines are spanned by scaled translates of a single B-spline. It provides an elementary strategy for hierarchical subdivision of B-splines. This technique is very easy to implement and well-suited for spline-based finite element approximations. Several types of weighted finite element subspaces are also described.
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