In time-dependent finite-element calculations, a mass matrix naturally arises. To avoid the solution of corresponding algebraic equation system at each time step, 'mass lumping' is widely used, even though this pragmatic diagonalization often reduces accuracy. We show how unassembled form equations can be used to establish (in an element-by-element manner) realistic upper and lower bounds on eigenvalues fully consistent when preconditioned by its diagonal entries. use technique give specific...

The computational solution of problems can be restricted by the availability methods for linear(ized) systems equations. In conjunction with iterative methods, preconditioning is often vital component in enabling such when dimension large. We attempt a broad review methods.

Standard Krylov subspace solvers for self-adjoint problems have rigorous convergence bounds based solely on eigenvalues. However, non-self-adjoint problems, eigenvalues do not determine behavior even widely used iterative methods. In this paper, we discuss time-dependent PDE which are always non-self-adjoint. We propose a block circulant preconditioner the all-at-once evolutionary system has Toeplitz structure. Through reordering of variables to obtain symmetric system, able rigorously...

Mixed finite element formulations give rise to large, sparse, block linear systems of equations, the solution which is often sought via a preconditioned iterative technique. In this work we present general analysis block-preconditioners based on stability conditions inherited from formulation method (the Babuska--Brezzi, or inf-sup, conditions). The motivated by notions norm-equivalence and field-of-values-equivalence matrices. particular, sufficient for diagonal triangular be norm-...

We consider the application of conjugate gradient method to solution large, symmetric indefinite linear systems. Special emphasis is put on use constraint preconditioners and a new factorization that can reduce number flops required by preconditioning step. Results concerning eigenvalues preconditioned matrix its minimum polynomial are given. Numerical experiments validate these conclusions.

Circulant preconditioning for symmetric Toeplitz linear systems is well established; theoretical guarantees of fast convergence the conjugate gradient method are descriptive seen in computations. This has led to robust and highly efficient solvers based on use Fourier transform exactly as originally envisaged [G. Strang, Stud. Appl. Math., 74 (1986), pp. 171--176]. For nonsymmetric systems, lack generally theory most iterative methods Krylov type provided a barrier such comprehensive...

McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40 (2018), pp. A1012–A1033] present a method for preconditioning time-dependent PDEs via an approximation by nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners the non-symmetric block Toeplitz which arise from all-at-once formulation. They suggest such approach might be efficiently implemented in parallel. In this short article, we parallel numerical results their preconditioner exhibit strong...

It is widely appreciated that the iterative solution of linear systems equations with large sparse matrices much easier when matrix symmetric. equally advantageous to employ symmetric methods a nonsymmetric self-adjoint in nonstandard inner product. Here, general conditions for such self-adjointness are considered. A number known examples saddle point surveyed and combined make new combination preconditioners which different products. In particular, method related Bramble–Pasciak CG...

SUMMARY Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the poses a significant additional challenge optimization methods. In this paper, we propose preconditioners saddle point that arise when primal–dual active set method is used. We also show same system can be derived considered as semismooth Newton method. addition, projected gradient employed to solve simple bounds, and discuss efficient...

When solving certain time-dependent partial differential equations using a finite-element technique on deforming grid, it is shown that there need to differentiate the trial solution with respect position of each moveable node points. A result presented which enables these nonstandard derivatives be expressed in terms standard spatial functions provided conditions are satisfied. These investigated and satisfied for large class spaces. Even when necessary not by all being used, still great...

The morning finite-element method for evolutionary partial differential equations leads to a coupled non-linear system of ordinary in time, with coefficien matrix A, say, the time derivaties, We show linear elements any number dimensions, A can be written form MTCM, where C depends solely on mesh geometry and M gradient section, As simple consequence we that is singular only cases (i) element degeneracy (⁠∣c∣=0⁠) (ii) collinearity nodes (M not out full rank). give constructions inversion all...