Finite Element Methods with Hierarchical WEB-splines

Piecewise polynomial approximations are fundamental to geometric modeling, computer graphics, and finite element methods. The classical finite element method uses low order piecewise polynomials defined on polygonal domains. The domains are discretized into simple polygons called the mesh. These polygons might be triangles, quadrilaterals, etc., for two-dimensional domains, and tetrahedra, hexahedra, etc., for three-dimensional domains. Meshing is often the most timeconsuming process in finite element methods. In classical two-dimensional finite element methods, the basis functions are usually hat functions defined on triangulations. Another possible selection of a finite element basis in two dimensions are tensor product b-splines. Bivariate B-splines are piecewise polynomials of degree n with support having (n+ 1) cells. The domain is discretized via a uniform grid. Relevant are those b-splines for which the support intersects the domain. To keep the support of a relevant B-spline within the domain, we multiply it by a weight function. The weight function is positive in the interior of the domain and vanishes on the boundary and outside of the domain. The resulting weighted B-splines conform to homogeneous boundary conditions. They satisfy the usual properties of a finite element basis. The insertion of new knots into the grid is not a good adaptive strategy because of the global effect of knot insertion. Instead, hierarchical refinement is very effective for tensor product splines. It permits the change of control points and subsequent editing of fine details in some parts while keeping the other parts unaffected. For programming, a data structure is required that not only keeps track of the refinement but also stores the information about the discretization of the domain. Moreover, algorithms for assembling and solving the finite element system are needed. In this thesis, we have developed such adaptive schemes with weighted B-splines and implemented them in Matlab with an appropriate data structure.