Solution of divergent type elliptic equations by a fourth order finite difference method (Q1907255)

A fourth-order finite difference approximation \(u_h\) to the Poisson equation \(\Delta u = f\) in a two-dimensional region \(\Omega\), \(u = g\) on \(\partial \Omega\), is found from solving \(\square_h = f + h^2 \Delta f\) in \(\Omega_h\), \(u_h = g\) on \(\partial \Omega_h\), where \(\square_h\) is the nine-point operator. The authors generalize this for the ``divergence'' type operator \(Lu \equiv \text{div}(p \text{ grad } u) = f\) in \(\Omega\), \(u = g\) on \(\partial \Omega\). They also consider the eigenvalue problem for \(L\) and generalizations when \(p\) and/or \(f\) depend on \(u\).