Discrete spectral problem with fourth order of precision (Q1317592)

In a commensurable rectangular region \(\Omega\), the eigenvalue problem \(Lu= \lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) is considered, where \[ L= {\partial\over \partial x_ 1} \left({1\over p_ 1} {\partial\over \partial x_ 1}\right)+ {\partial\over \partial x_ 2} \left({1\over p_ 2} {\partial\over \partial x_ 2}\right), \] and where the smooth coefficients satisfy \(0< c_ 1\leq p_ j(x_ 1,x_ 2)\leq c_ 2\) \((j=1,2)\). After introducing a square grid of meshsize \(h\), the usual second order discretization \(A^ 0\) of \(L\) is perturbed to yield \(A^ 0+ (h^ 2/12)A^*\). Here, \(A^*\) is the discretization of a special fourth order differential operator \(L^*\) which is (formally) selfadjoint and positiv definite under an additional boundary condition. The perturbed scheme is shown to be of order four.