A generalized SSOR method

Abstract

To solve large sparse systems of linear equations with symmetric positive definite matrixA=D+L+L*,D=diag(A), with iteration, the SSOR method with one relaxation parameter ω has been applied, yielding a spectral condition number approximately equal to the square root of that ofA, if the condition$S(\tilde L\tilde L*) \leqq \tfrac{1}{4}$, where$\tilde L = D^{ - \tfrac{1}{2}} LD^{ - \tfrac{1}{2}}$, is satisfied and if 0<ω<2 is chosen optimally. The matrix arising from the differenced Dirichlet problem satisfies in general the spectral radius condition given above, only if the coefficients of the differential equation are constant and if the mesh-width is uniform.

However, using one relaxation parameter for each mesh-point, the main result for the SSOR method, that the spectral condition number varies with a parameterζ > 0 likeO([ζ ⁻¹ +ζ/λ ₁]h ⁻¹),h → 0, whereλ ₁ h ² is the smallest eigenvalue ofD ⁻¹ A, carries over for variable smooth coefficients and even for certain kinds of discontinuities among the coefficients, if the mesh-width is adjusted properly in accordance with the discontinuity.

Since the resulting matrix of iteration has positive eigenvalues, a semi-iterative technique can be used. The necessary number of iterations is thus onlyO(h ^−1/2).