Adaptive BDDC Preconditioner For Two Types Of Discrete Systems Of Partial Differential Equations

Helmholtz equation and diffusion equation are two important mathematical models.The former is the basic model to simulate the electromagnetic radiation,acoustic scattering,exploration seismology and other engineering problems.The latter is widely used in scientific and engineering computation,such as multi-fluid hydrodynamic,fluid-solid coupling mechanics and so on.Research on efficient nu?merical algorithms for solving Helmholtz equation with high wave number and the complex diffusion equation is an important topic in the field of scientific engineer-ing computing nowadays.Adaptive BDDC algorithm is a class of nonoverlapping domain decomposition method developed in recent years.How to design and an-alyze the corresponding efficient adaptive BDDC preconditioners for the systems arising from these two kinds of PDEs is worth studying.This paper focuses on this work,and the main results are as follows.For the abstract variational problem whose Schur complement system is Her-mitian positive definite,we describe the function space which the adaptive BDDC preconditoners rely on by introducing the basis functions of several auxiliary spaces and an inequalities which the dual-primal basis functions satisfied on each inter-face,and arrive at the framework of constructing adaptive two-level BDDC pre-conditioner in variational form.Furthermore,by deriving the control inequality that two local functions satisfy,we establish the framework of theoretical analysis for the adaptive two-level BDDC preconditioned system in variational form and prove that the condition number is bounded above by C?,where the constant C is only dependent on the maximum number of edges per subdomain,and ? is a given tolerance.In addition,a multilevel adaptive BDDC algorithm is designed to resolve the bottleneck for solving large scale coarse problem.For the weighted plane wave least-squares(PWLS)discretization of Helmholtz equation with high wave number,an adaptive BDDC preconditioner for solving the corresponding Schur complement system is constructed and analyzed under the framework of adaptive BDDC method mentioned above.We introduce a spe-cial "interface" and some local techniques during constructing the corresponding adaptive BDDC preconditioner,since the unknowns are defined on the elements rather than vertices or edges which is different from the traditional finite element method,and the condition number of PWLS discretization is highly sensitive to the unit degree of freedom.Numerical results show that the iterations counts of the adaptive two-level BDDC preconditioner is weakly dependent on the angular frequency,the number of subdomains and the size of the grid.Especially,the ex-periments on a singular example show that the two-level algorithm is also robust to the PWLS discrete system for solving Helmholtz equation with complex wave number,and the multilevel algorithm can effectively reduce the scale of the coarse space,and is effective for the Helmholtz problem with high wave number.For high-order mortar discretization of diffusion equations with high varying and random coefficients,an adaptive two-level BDDC preconditioner is developed and analyzed under the framework of adaptive BDDC method.Compared with the adaptive BDDC algorithms for conforming Galerkin approximations,our algorithm is simpler,because there have not any continuity constraints at subdomain vertices in the mortar method involved in this paper.Numerical results show the robustness and efficiency of the algorithm for various models whether the subdomains are geometric matching or not.For diffusion problems with strong discontinuity and multiple material cells on the Eulerian quadrilateral grids,we present a vertex-centered MACH-like finite volume method by introducing different ways of averaging for the coefficient of hybrid element.This method is motivated by Frese[No.AMRC-R-874,Mission Research Corp.,Albuquerque,NM,1987].Then for the degenerate five-point MACH-like scheme,we derived different local truncation errors under different assumptions,especially on the interior nodes neighbouring the interface,the local truncation errors are O(h)and O(h2)respectively,and we also get the global error estimates O(h| ln h|)by introducing the decomposition of the error difference equation,dimension-reduction method based on discrete sine transform and some new analysis techniques.Furthermore,we prove that this scheme can reach the asymptotic optimal error estimate O(h2| ln h|)in the maximum norm under some assumptions.Since the degree of freedom in the MACH-like scheme is defined on the grid vertices,similar to the adaptive BDDC preconditioner for the conforming Galerkin method,we constructed an adaptive BDDC algorithm for solving the Schur complement system of the MACH-like FVM.Numerical experiments verify the theoretical results and the robustness of preconditioner.