GenEO ('Generalised Eigenvalue problems on the Overlap') is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers which are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection-diffusion-reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalised eigenvalue problem based on a self-adjoint and positive-definite subproblem. We obtain GMRES iteration counts which are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on the variation of the other coefficients. While the iteration number estimates do grow as the non-self-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus we obtain an iterative solver which is efficient in parallel and very effective for a wide range of convection-diffusion-reaction problems.
翻译:Geneo(Geneo)是计算一个操作者依赖的光谱偏重值问题的一种方法,该方法将计算一个操作者依赖的光谱偏小空间,并与子域的本地解决方案相结合,形成一个对流派PDE的强大平行的平行地分解先决条件。以前,在自对接和正分解的情况下,已经证明这种方法,当用作调和梯度的前提条件时,产生与部分差分操作者系数领域差异完全独立的迭代数。我们将这一理论推广到对流-分散-反应问题的当地解决方案,以形成一个对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-对流-和对立-和对立-对立-对立-对等-和对立-对立-对流-对流-对流-对流-对流-对立-对流-和对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-)-对等-对等-、对等-对等-对等-对等-对等-对等-、对等-对等-对等-对调-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-对等-