Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in $\mathbb {R}^3$

Abstract

The aim of this paper is to present spline methods for the numerical solution of integral equations on surfaces of $\mathbb {R}^3$, by using optimal superconvergent quasi-interpolants defined on type-2 triangulations and based on the Zwart–Powell quadratic box spline. In particular we propose a modified version of the classical collocation method and two spline collocation methods with high order of convergence. We also deal with the problem of approximating the surface. Finally, we study the approximation error of the above methods together with their iterated versions and we provide some numerical tests.

Appendix A

Here we report the expression of the fundamental functions associated with $Q_{mn}$ defined in (2.1), for $m,n \ge 8$. They are obtained from the coefficient functionals given in [16]. For the pairs (i, j) with $i=4,\ldots ,m-3$ and $j=4,\ldots ,n-3$

$$\begin{aligned} L_{i,j}=\frac{3}{2}B_{i,j}-\frac{1}{8}(B_{i,j-1}+B_{i,j+1}+B_{i-1,j}+B_{i+1,j}). \end{aligned}$$

The other $L_{i,j}$’s have particular definitions. In the neighbourhood of the point (a, c) we have

$$\begin{aligned} \begin{array}{ll} L_{0,0}=&{}\frac{1403}{504}B_{0,0}-\frac{4}{15}B_{1,1}, \\ &{}\\ L_{1,0}=&{}\frac{131}{60}B_{1,0}-\frac{173}{300}B_{0,1}-\frac{1}{12}B_{2,1},\\ &{}\\ L_{2,0}=&{}-\frac{397}{1440}B_{0,0}-\frac{2}{15}B_{1,1}+\frac{12}{5}B_{2,0}-\frac{1}{12}B_{3,1}-\frac{7}{30}B_{2,1}+\frac{9}{40}B_{1,0}, \\ &{}\\ L_{3,0}=&{}-\frac{1}{12}B_{4,1}+\frac{3}{20}B_{0,1}+\frac{12}{5}B_{3,0}-\frac{1}{12}B_{2,1}-\frac{7}{30}B_{3,1}, \\ &{}\\ L_{4,0}=&{}\frac{11}{224}B_{0,0}+\frac{12}{5}B_{4,0}-\frac{1}{120}B_{1,0}-\frac{7}{30}B_{4,1}-\frac{1}{12}B_{3,1}-\frac{1}{12}B_{5,1}, \\ &{}\\ \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{ll} L_{1,1}=&{}-\frac{63}{32}B_{0,0}-\frac{13}{40}(B_{1,0}+B_{0,1})+\frac{33}{20}B_{1,1}-\frac{1}{4}(B_{2,0}+B_{0,2}), \\ &{}\\ L_{2,1}=&{}-\frac{47}{60}B_{1,0}-\frac{9}{8}B_{2,0}-\frac{1}{4}B_{3,0}-\frac{1}{20}B_{1,1}+\frac{13}{8}B_{2,1}+\frac{1}{8}B_{0,2}-\frac{1}{24}B_{1,2}-\frac{1}{8}B_{2,2}, \\ &{}\\ L_{3,1}=&{}\frac{3}{50}B_{1,0}-\frac{1}{4}B_{2,0}-\frac{9}{8}B_{3,0}-\frac{1}{4}B_{4,0}-\frac{7}{40}B_{0,1}+\frac{1}{40}B_{1,1}+\frac{13}{8}B_{3,1}-\frac{1}{8}B_{3,2}, \\ &{}\\ L_{2,2}=&{}\frac{317}{288}B_{0,0}+\frac{1}{4}(B_{0,1}+B_{1,0})+\frac{1}{8}(B_{3,0}+B_{0,3})-\frac{1}{15}B_{1,1}-\frac{1}{6}(B_{1,2}+B_{2,1})\\ &{}-\frac{1}{24}(B_{1,3}+B_{3,1})-\frac{1}{8}(B_{2,3}+B_{3,2})+\frac{3}{2}B_{2,2},\\ &{}\\ L_{3,2}=&{}-\frac{37}{160}B_{0,0}+\frac{1}{8}B_{2,0}+\frac{1}{8}B_{4,0}-\frac{1}{24}B_{2,1}-\frac{1}{24}B_{4,1}-\frac{1}{6}B_{3,1}-\frac{1}{40}B_{0,2}\\ &{}+\frac{1}{40}B_{1,2}-\frac{1}{8}B_{2,2}-\frac{1}{8}B_{4,2}+\frac{3}{2}B_{3,2}-\frac{1}{8}B_{3,3}, \\ &{}\\ L_{3,3}=&{}-\frac{1}{40}(B_{3,0}+B_{0,3})+\frac{1}{40}(B_{3,1}+B_{1,3})-\frac{1}{8}(B_{3,2}+B_{2,3})+\frac{3}{2}B_{3,3}\\ &{}-\frac{1}{8}(B_{3,4}+B_{4,3}). \end{array} \end{aligned}$$

Along the lower edge, for $i=5,\ldots ,m-4$, we have:

$$\begin{aligned} \begin{array}{ll} L_{i,0}=&\frac{12}{5}B_{i,0}-\frac{7}{30}B_{i,1}-\frac{1}{12}(B_{i-1,1}+B_{i+1,1}), \end{array} \end{aligned}$$

and for $i=4,\ldots ,m-3$:

$$\begin{aligned} \begin{array}{ll} L_{i,1}=&{}-\frac{9}{8}B_{i,0}-\frac{1}{4}(B_{i-1,0} +B_{i+1,0})+\frac{13}{8}B_{i,1}-\frac{1}{8}B_{i,2}, \\ &{}\\ L_{i,2}=&{}\frac{1}{8}(B_{i-1,0}+B_{i+1,0})-\frac{1}{6} B_{i,1}-\frac{1}{24}(B_{i-1,1}+B_{i+1,1})+\frac{3}{2}B_{i,2}\\ &{}-\frac{1}{8}(B_{i-1,2}+B_{i+1,2})-\frac{1}{8}B_{i,3},\\ &{}\\ L_{i,3}=&{}-\frac{1}{40}B_{i,0}+\frac{1}{40}B_{i,1}-\frac{1}{8}B_{i,2}+\frac{3}{2}B_{i,3}-\frac{1}{8}(B_{i-1,3}+B_{i+1,3})-\frac{1}{8}B_{i,4}.\\ \end{array} \end{aligned}$$

Taking into account the coefficient functional symmetries, analogous formulas exist for the three other edges and vertices of $\varOmega$.

We remark that in case $m,n < 8$ the fundamental functions have particular expressions, always obtained from the coefficient functionals given in [16].