Adaptive quadratures over volumes (Q1180343)

An adaptive extrapolative quadrature algorithm is presented for approximation of volume integrals \(\int_ Df dx\) over a region \(D\) in \(\mathbb{R}^ 3\) with piecewise smooth boundary. \(f(x)\) may have a weak singularity inside \(D\). The algorithm uses the Coxeter-Freudenthal triangulation method to subdivide \(D\). Briefly the procedure is the following: Having generated a subdivision (tetrahedron) \(\sigma\) in \(D\) the algorithm either (i) generates \(\int_ \sigma f dx\) to within a prescribed tolerance and advances to the next subdivision or (ii) partitions \(\sigma\) into smaller tetrahedrons by halving the edges and applies (i), (ii) to these new subdivisions. Procedure (i) involves an extrapolation using a Romberg extrapolation tableaux and it employs different quadrature formulas depending on how many of the vertices of the tetrahedron cross the boundary of \(D\). Numerical results are given for several test problems obtained with a program written in Turbo-C and run on a PC with 80386/387 processors. A high degree of accuracy was obtained.