Owe Axelsson

A comparison of iterative methods to solve complex valued linear algebraic systems

Complex valued linear algebraic systems arise in many important applications. We present analytical and extensive numerical comparisons of some available numerical solution methods. It is advocated, in particular for large scale ill-conditioned problems, to rewrite the complex-valued system in real valued form leading to a two-by-two block system of particular form, for which it is shown that a very efficient and robust preconditioned iterative solution method can be constructed. Alternatively, in many cases it turns out that a simple preconditioner in the form of the sum of the real and the imaginary part of the matrix also works well but involves complex arithmetic.

Numerical and computational efficiency of solvers for two-phase problems

We consider two-phase flow problems, modelled by the Cahn-Hilliard equation. In this work, the nonlinear fourth-order equation is decomposed into a system of two coupled second-order equations for the concentration and the chemical potential. We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning technique that reduces the problem of solving the non-symmetric discrete Cahn-Hilliard system to a problem of solving systems with symmetric positive definite matrices where off-the-shelf multilevel and multigrid algorithms are directly applicable. The resulting solution methods exhibit optimal convergence and computational complexity properties and are suitable for parallel implementation. We illustrate the efficiency of the proposed methods by various numerical experiments, including parallel results for large scale three dimensional problems.

Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices

Poroelastic models arise in reservoir modeling and many other important applications. Under certain assumptions, they involve a time-dependent coupled system consisting of Navier–Lamé equations for the displacements, Darcy’s flow equation for the fluid velocity and a divergence constraint equation. Stability for infinite time of the continuous problem and, second and third order accurate, time discretized equations are shown. Methods to handle the lack of regularity at initial times are discussed and illustrated numerically. After discretization, at each time step this leads to a block matrix system in saddle point form. Mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented. A certain block matrix preconditioner is shown to cluster the eigenvalues of the preconditioned matrix about the unit value but needs inner iterations for certain matrix blocks. The strong clustering leads to very few outer iterations. Various approaches to construct preconditioners are presented and compared. The sensitivity of the number of outer iterations to the stopping accuracy of the inner iterations is illustrated numerically.

On the solution of high order stable time integration methods

Evolution equations arise in many important practical problems. They are frequently stiff, i.e. involves fast, mostly exponentially, decreasing and/or oscillating components. To handle such problems, one must use proper forms of implicit numerical time-integration methods. In this paper, we consider two methods of high order of accuracy, one for parabolic problems and the other for hyperbolic type of problems. For parabolic problems, it is shown how the solution rapidly approaches the stationary solution. It is also shown how the arising quadratic polynomial algebraic systems can be solved efficiently by iteration and use of a proper preconditioner.

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

High order time discretization for DAEs with efficient block preconditioners

The contribution considers parabolic PDEs describing uniphysics problems like nonstationary Darcy flow and their extension to multiphysics like poroelasticity problems. Discretization is assumed by mixed and standard/mixed finite elements in space and stable higher order methods in time. Parallelizable preconditioners for iterative solution of linear systems arising within the time steps are suggested and analysed. The analysis shows that the preconditioned systems are diagonalizable with very localized spectra. It indicates possible very fast convergence of Krylov type methods, which was also confirmed by numerical experiments with two step Radau time integration method.

Continuation Newton methods

Severely nonlinear problems can only be solved by some homotopy continuation method. An example of a homotopy method is the continuous Newton method which, however, must be discretized which leads to the damped step version of Newtons method.The standard Newton iteration method for solving systems of nonlinear equations F ( u ) = 0 must be modified in order to get global convergence, i.e. convergence from any initial point. The control of steplengths in the damped step Newton method can lead to many small steps and slow convergence. Furthermore, the applicability of the method is restricted in as much as it assumes a nonsingular and everywhere differentiable mapping F ( ? ) .Classical continuation methods are surveyed. Then a new method in the form of a coupled Newton and load increment method is presented and shown to have a global convergence already from the start and second order of accuracy with respect to the load increment step and with less restrictive regularity assumptions than for the standard Newton method. The method is applied for an elastoplastic problem with hardening.

Reaching the superlinear convergence phase of the CG method

The rate of convergence of the conjugate gradient method takes place in essentially three phases, with respectively a sublinear, a linear and a superlinear rate. The paper examines when the superlinear phase is reached. To do this, two methods are used. One is based on the K -condition number, thereby separating the eigenvalues in three sets: small and large outliers and intermediate eigenvalues. The other is based on annihilating polynomials for the eigenvalues and, assuming various analytical distributions of them, thereby using certain refined estimates. The results are illustrated for some typical distributions of eigenvalues and with some numerical tests.

Preconditioners for Some Matrices of Two-by-Two Block Form, with Applications, I

Matrices of two-by-two block form with matrix blocks of equal order arise in various important applications, such as when solving complex-valued systems in real arithmetics, in linearized forms of the Cahn–Hilliard diffusive phase-field differential equation model and in constrained partial differential equations with distributed control. It is shown how an efficient preconditioner can be constructed which, under certain conditions, has a resulting spectral condition number of about 2. The preconditioner avoids the use of Schur complement matrices and needs only solutions with matrices that are linear combinations of the matrices appearing in each block row of the given matrix and for which often efficient preconditioners are already available.

Macro-elementwise preconditioning methods

Extremely large scale problems, modelled by partial differential equations, arise in various applications, and must be solved by properly preconditioned iterative methods. Frequently, the corresponding medium is heterogeneous. Recursively constructed two-by-two block matrix partitioning methods and elementwise constructed preconditioners for the arising pivot block and Schur complement matrices have turned out to be very efficient methods, and are analysed in this paper. Thereby special attention is paid to macroelementwise partitionings, which can be particularly efficient in the modelling of materials with large and narrow variations and can also provide efficient implementations on parallel computers.