Sixth order compact finite difference schemes for Poisson interface problems with singular sources

A High Order Compact Finite Difference Scheme for Elliptic Interface Problems with Discontinuous and High-Contrast Coefficients

“…where all c k, ,i and c f,m,n,j are to-be-determined constants. Similar to [9], we observe that the coefficients of a compact scheme are nontrivial if C k, (0) = 0 for at least some k, = −1, 0, 1, that is, c k, ,0 = 0 for at least some k, = −1, 0, 1. Similar to Eq.…”

“…Since the function u is a solution for the partial differential equation in (1.1), all the quantities u (m,n) , (m, n) ∈ Λ M +1 are not independent of each other. Similar to the Lemma 2.1 in [9], we have the following result:…”

“…Similarly as the discussion for the irregular points in [9], the identities in (2.5) and (2.13) hold by replacing a, u and f by a ± , u ± and f ± , i.e.,…”

“…In [9], we derived a sixth order compact finite difference scheme for the Poisson equation with singular sources, whose solution has a discontinuity across a smooth interface. The most important feature of the scheme is that the matrix of the resulting linear system is independent of the location of the singularity in the source term.…”

“…By (2.9) and (2.10) in [9] and (2.5) of this paper, the approximation of u(x + x * i , y + y * j ) in (2.4) can be written as…”