A rapidly convergent method for the inversion of separable, positive, self-adjoint discrete elliptic operators in three or more dimensions
The authors observe in this brief note that if the operators \(L_ x\), \(L_ y\), \(L_ z\) have positive eigenvalues and commute, then the problem \((L_ x+L_ y+L_ z)\phi =\rho\) can be solved by the iteration \((L_ x+L_ y)\phi^{k+1}=\rho -L_ z\phi \quad k,\) \((L_ y+L_ z)\phi^{k+2}=\rho -L_ x\phi^{k+1},\quad (L_ z+L_ x)\phi^{k+3}=\rho -L_ y\phi^{k+2}.\) Analysis of each component of the eigenfunction expansion shows \(| \phi^{k+3}-\phi \quad k| \leq | \phi \quad k-\phi^{k-3}| /8.\) The authors remark that the case of a zero eigenvalue can be handled, and also that the method can be used as an inner iteration in an obvious extension to four dimensions and higher.
separable, positive, self-adjoint discrete elliptic operators
scientific article; zbMATH DE number 4054936
scientific article