Contour Integral Solution of Elliptic PDEs in Cylindrical Domains

The solutions of certain elliptic PDEs can be expressed as contour integrals of Dunford type. In this paper efficient contours and quadrature rules for the approximation of such integrals are proposed. The trapezoidal and midpoint rules are used in combination with a conformal mapping that fully exploits the analyticity of the integrand, leading to rapidly converging quadrature formulas of double exponential type. In addition to optimizing the step size of the quadrature formula, implementation aspects such as the solution of the resulting shifted linear systems at each quadrature node are discussed. Numerical examples involving Laplace’s equation in a rectangle, a box, and an annular cylinder are presented. Timing and accuracy comparisons of the various implementations are given.