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Finally, since w is continuous, the boundary of DI n D2 is correctly represented as {x : w (jc) = 0}. 3). Hence, the standard R-function method yields Even for this simple example, the form of the weight function is quite complicated. But, of course, the explicit formula is not needed. All computations can be performed by successively evaluating the expressions involved, similarly as in the processing of tree structures in CAD models. Moreover, derivatives of w can be evaluated with the aid of automatic differentiation [40, 69].

As an example, we apply the Laplace operator A = ]Py 3^ to a multivariate B-spline. 2 yields and A£>£ h is the sum of these expressions over all unit vectors a. As a further application, we compute the integrals which appear in the Ritz-Galerkin approximation of Poisson's equation. , Chapter 4. 2. Ritz-Galerkin integrals g0^, \tv\ < 2, for biquadratic B-splines. 12 of the previous section. 2 shows the integrals for biquadratic B-splines. Since gk,i depends only on the difference k — t, we have chosen k — 0 and associated each value with the center of the support ofblh.

2 since b°(x — y) equals 1 for y e (x — 1, ;c] and vanishes outside this interval. The general case follows by induction on m. Since both sides vanish for jc = 0, differentiating yields the equivalent equation which is valid by induction hypothesis. 11 easily yields a formula for the scalar product of two B-splines bnkh and b" h. Substituting y = z/h — t, dz — hdy, the argument of the first B-spline becomes y + t — k. ). This proves the first part of the following theorem. 12 Scalar Products The scalar products of the B-splines b^h,bnih and of their derivatives are respectively.