Overlapping Schwarz methods for mixed linear elasticity and Stokes problems

In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries. Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only e2-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the e2-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization. In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows NJ on that level. Finally, numerical results are provided.

A new fast method to compute saddle-points in constrained optimization and applications

A dual iterative substructuring method with a penalty term in three dimensions

In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems, including the Stokes equations for viscous incompressible flow, mixed formulation of diffusion equations for groundwater flow, time-harmonic Maxwell equations for electromagnetics, etc. Due to the lack of knowledge or intrinsic randomness, the (viscosity, diffusivity, permeability, permittivity, etc.) coefficients of such problems are uncertain and can often be represented or approximated by high- or countably infinite-dimensional random parameters equipped with suitable probability distributions, and the coefficients affinely depend on a series of either globally or locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, or adaptive wavelet approximations. We consider sparse polynomial approximations of the parametric solutions, in particular sparse Taylor approximations, and study their convergence properties for these parametric problems. Under suitable sparsity assumptions on the parametrization of the random coefficients, we show the algebraic convergence rates O(N−r) for the sparse polynomial approximations of the parametric solutions based on the results for affine parametric elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., et al.: ESAIM Math. Model. Numer. Anal. 51, 321–339, 2017), (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015), (Chkifa, A., et al.: J. Math. Pures Appl. 103, 400–428, 2015), (Chkifa, A., et al.: ESAIM Math. Model. Numer. Anal. 47, 253–280, 2013), (Cohen, A., Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), with the rate r depending only on a sparsity parameter in the parametrization, not on the number of active parameter dimensions or the number of polynomial terms N. We note that parametric saddle point problems were considered in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2) with the anticipation that the same results on the approximation of the solution map obtained for elliptic PDEs can be extended to more general saddle point problems. In this paper, we consider a general formulation of saddle point problems, different from that presented in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2), and obtain convergence rates for the two variables, e.g., velocity and pressure in Stokes equations, which are different for the case of locally supported basis functions.

In this work, we consider numerical methods for solving a class of block three-by-three saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and a class of quadratic programs. We present a variant of Uzawa method with two variable parameters for the saddle point problems. These two parameters can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. We show that the new iterative method converges to the unique solution of the saddle point problems under a reasonable condition. Numerical experiments highlighting the performance of the proposed method for problems are presented.

Saddle point problems appear in many scientific and engineering applications. Hence, it is important to solve them fast with high accuracy. We have proposed the method for saddle point problems using block structure in order to solve them fast. In our proposed method, first, a linear system with multiple right-hand sides is solved instead of solving the saddle point problems directly. After that, a small linear system with a dense matrix is solved. In our previous work, it has been observed that our method is faster than the conventional approach, but the accuracy of the obtained approximate solutions is worse than the conventional one. In this paper, we propose to improve the accuracy of the approximate solutions of the saddle point problems by improving the accuracy of the solution of the small linear system using the mixed precision iterative refinement technique. Numerical experiments illustrate that the proposed approach improves the accuracy of the approximate solutions to the same extent as the conventional approach.

An iterative substructuring method with Lagrange multipliers is considered for second order elliptic problems, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with the saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter η, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. In spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of η, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. Computational issues and numerical results are presented.

Iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems. An optimal preconditioner which is independent of the discretization and the penalty parameter is constructed. This approach is then used to design an iterative method with a convergence rate bounded independently of the Lamé parameters occurring in the equations of linear elasticity.

This article presents a general and novel approach to the automation of goal-oriented error control in the solution of nonlinear stationary finite element variational problems. The approach is based on automated linearization to obtain the linearized dual problem, automated derivation and evaluation of a posteriori error estimates, and automated adaptive mesh refinement to control the error in a given goal functional to within a given tolerance. Numerical examples representing a variety of different discretizations of linear and nonlinear partial differential equations are presented, including Poisson's equation, a mixed formulation of linear elasticity, and the incompressible Navier--Stokes equations.

The $Q^N$- $Q^{N-2}$ spectral element discretization of the Stokes equation gives rise to an ill-conditioned, indefinite, symmetric linear system for the velocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original system is reduced to a symmetric equation for the velocity, which can be solved with the conjugate gradient method. We prove that the condition number of the iteration operator is bounded from above by $C(1+\log(N))^3/\beta_N$, where C is a positive constant independent of the degree N and the number of subdomains, and $\beta_N$ is the inf-sup condition of the pair $Q^N$- $Q^{N-2}$. We also consider the stationary Navier-Stokes equations; in each Newton step, a non-symmetric indefinite problem is solved using a Schwarz preconditioner. By using an especially designed low-order global space, and the solution of local problems analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the method. We prove that the number of iterations of the GMRES method, at each Newton step, is bounded from above by $C(1+\log(N))^3/\beta_N$. The constant C does not depend on the number of subdomains or N, and it does not deteriorate as the Newton iteration proceeds.

A posteriori error analysis for two non-overlapping domain decomposition techniques

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of l2-norms of wavelet expansion coefficients and the constraints are written in form of an l2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in l2. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

We discuss a class of preconditioning methods for the iterative solution of symmetric algebraic saddle point problems, where the (1, 1) block matrix may be indefinite or singular. Such problems may arise, e.g. from discrete approximations of certain partial differential equations, such as the Maxwell time harmonic equations. We prove that, under mild assumptions on the underlying problem, a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) can be chosen in a way which guarantees that the convergence rate of the preconditioned conjugate residuals method is independent of the discretization mesh parameter. We provide examples of such preconditioners that do not require additional scaling. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper we derive bounds on the eigenvalues of the preconditioned matrix that arises in the solution of saddle point problems when the Hermitian and skew-Hermitian splitting preconditioner is employed. We also give sufficient conditions for the eigenvalues to be real. A few numerical experiments are used to illustrate the quality of the bounds.