Abstract
We describe a collocation method with weighted extended B-splines
(WEB-splines) for arbitrary bounded multidimensional domains,
considering Poisson's equation as a typical model problem. By slightly
modifying the B-spline classification for the WEB-basis, the centers of
the supports of inner B-splines can be used as collocation points. This
resolves the mismatch between the number of basis functions and
interpolation conditions, already present in classical univariate
schemes, in a simple fashion. Collocation with WEB-splines is
particularly easy to implement when the domain boundary can be
represented as zero set of a weight function; sample programs are
provided on the website http://www.web-spline.de. In contrast to
standard finite element methods, no mesh generation and numerical
integration is required, regardless of the geometric shape of the
domain. As a consequence, the system equations can be compiled very
efficiently. Moreover, numerical tests confirm that increasing the
B-spline degree yields highly accurate approximations already on
relatively coarse grids. Compared with Ritz-Galerkin methods, the
observed convergence rates are decreased by 1 or 2 when using splines of
odd or even order, respectively. This drawback, however, is outweighed
by a substantially smaller bandwidth of collocation matrices.
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