Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems

. A new approach of constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners are then constructed for this system based on a triangulation of parallelepipeds into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory. Key words. mixed method, nonconforming method, multilevel preconditioner, condition number, second order elliptic problem AMS(MOS) subject classifications. 65N30, 65N22..

Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems

this paper is construction, study, and testing of optimal (or almost optimal) preconditioners for the algebraic problem (1.3) with symmetric full tensors a(x) and possibly large jumps and anisotropy. Our analysis is restricted to domains which are topologically equivalent to parallelepipeds or are unions of such subdomains. The proposed construction uses the hybridized version of (1.3) and its equivalence to certain modifications of the nonconforming Galerkin approximation to (1.1) (see, e.g., [1,2,11,12,27]). Below we explain our approach in detail. Since V h ae V the vector functions in V h have normal components which are continuous across the interelement boundaries. Following [2], we relax this constraint on V h by defining ~ V h = fv 2

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

Iterative solution of saddle point problems using divergence-free finite elements with applications to groundwater flow

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