|Peter Alfeld||Department of Mathematics||College of Science||University of Utah|
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|Spline Spaces and Minimal Determining Sets||User's Guide||Residual Arithmetic|
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We illustrate the use of the applet on this page with several examples. Brief descriptions are given here, for details click on the relevant head line.
The greatest obstacle in multivariate spline research is the fact that the dimension of spline spaces (and also for example the solvebility of interpolation problems and the composition of minimal determining sets) depends on not just the topolgy of the triangulation but also its >em>geometry. An arbitrarily small change in the location of the vertices (without changing the way triangles are connected) can change the dimension of a spline space! This is in complete contrast to to the univariate case (of splines in one variable defined on a partition of an interval).
The case of r=1, d=3 is of particular interest for applications, and it leads to some famous and difficult open problems.
One of the fundamental aspects of multivariate splines is that things get easier as the polynomial degree increases. In particular, if d > 4r+1 a minimal determining set can be found by considering vertices, edges, and interiors of triangles individually.
Finite elements are special cases of multivariate splines. Their key attribute is that on each triangle they can be defined completely in terms of data or parameters given on that triangle. They are usual used for the solution of differential equations in which case the coefficients are unknowns that are found by assembling and solving a larger linear system. Finite elements being local causes this system to be (sparse
When the function and derivative values are given one can use finite element for interpolation, and that is how they will be presented on this page.