Suppose that \(P_{h_x,h_y}(u)\) represents the solution operator of a boundary value problem for an elliptic partial differential equation on the unit square broken into a grid of uniform rectangles of size \(h_x\times h_y\). The combination solution for \(h=2^{-n}\) is defined by \[ u_h^c=\sum_{i=1}^nP_{2^{-i},2^{i-n-1}}(u)- \sum_{i=1}^{n-1}P_{2^{-i},2^{i-n}}(u). \] Clearly, \(u_h^c\) can be computed in very few operations and without loss of efficiency on a parallel computers. Furthermore, its computational advantage improves in higher dimensions. The disadvantage of the method is that convergence proofs are not yet available for a large class of common problems. The current paper extends previous results and introduces new techniques of proof.
Among the several theorems concerning boundary problems with variable coefficients in two dimensions, the weakest of the authors’ results is that if \(u\) is the solution to such a problem, then \[ \|u-u_h^c\|_{H^1}\leq h\log(1/h) \|u\| \] under suitable assumptions on the coefficients and with suitable choice of norm on the right. The authors prove a similar estimate for combination solutions of the Poisson equation in higher dimensions. These theorems are generalizations of earlier results in that they weaken both the conditions on the coefficients and also lead to further generalizations to curvilinear mapped mesh grids.
The basic idea behind the proofs is to employ is the standard finite element operator using bilinear elements on rectangles as solution operator \(P\). The behavior of errors in combinations of mesh (axis) directions is well known and it is possible to apply superconvergence results to achieve the desired error estimates.