Solving Poisson-type equations with Robin boundary conditions on piecewise smooth interfaces

Highlights: • Two second-order accurate finite volume schemes for elliptic equations subject to Robin boundary conditions. • Irregular domains with piecewise smooth boundaries. • First scheme is symmetric and the other one is nonsymmetric with first- and second-order accurate gradients, respectively. • A hierarchical geometric reconstruction algorithm for integration over implicitly defined piecewise smooth geometries. • Numerical examples in two and three spatial dimensions. -- Abstract: We present two finite volume schemes to solve a class of Poisson-type equations subject to Robin boundary conditions in irregular domains with piecewise smooth boundaries. The first scheme results in a symmetric linear system and produces second-order accurate numerical solutions with first-order accurate gradients in the $L^{\infty }$-norm (for solutions with two bounded derivatives). The second scheme is nonsymmetric but produces second-order accurate numerical solutions as well as second-order accurate gradients in the $L^{\infty }$-norm (for solutions with three bounded derivatives). Numerical examples are given in two and three spatial dimensions.