Arbitrary order monotonic finite-volume schemes for 2D elliptic problems

Abstract

Monotonicity is very important in most applications solving elliptic problems. Many schemes preserving positivity has been proposed but are at most second-order convergent. Besides, in general, high-order schemes do not preserve positivity. In the present paper, we propose an arbitrary-order monotonic method for elliptic problems in 2D. We show how to adapt our method to the case of a discontinuous and/or tensorvalued diffusion coefficient, while keeping the order of convergence. We assess the new scheme on several test problems