arXiv subjects

Jan Papež

Publications and source records attributed to Jan Papež.

5 recordsLinked to original sources

arXiv2026

A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems

In this work, we develop algebraic solvers for linear systems arising from the discretization of second-order elliptic partial differential equations by saddle-point mixed finite element methods of arbitrary polynomial degree $p \ge 0$ on possibly highly graded simplicial meshes. We present a multigrid and a two-level domain decomposition approach in two and three space dimensions, steered by a posteriori estimators of the algebraic error. First, we extend [Miraçi, Papež, and Vohralík, SIAM J. Sci. Comput. 43 (2021), S117-S145] to the mixed finite element setting. Extending the multigrid procedure itself is rather natural. To obtain analogous theoretical results, however, a $p$-robust multilevel stable decomposition of the velocity space is needed. In two space dimensions, we can treat the velocity space as the curl of a stream-function Lagrange space, for which the previous results apply. In three space dimensions, we design a novel stable decomposition by combining a one-level high-order local stable decomposition of [Falk and Winther, Found. Comput. Math. (2025), DOI 10.1007/s10208- 025-09700-2] and a multilevel lowest-order stable decomposition of [Hiptmair, Wu, and Zheng, Numer. Math. Theory Methods Appl. 5 (2012), 297-332]. This allows us to prove that our multigrid solver contracts the algebraic error at each iteration $p$-robustly and, simultaneously, that the associated a posteriori estimator is $p$-robustly efficient. Next, we use this multilevel methodology to define a two-level domain decomposition method where the subdomains consist of overlapping patches of coarse-level elements sharing a common coarse-level vertex. We again establish a $p$-robust contraction of the solver and $p$-robust efficiency of the a posteriori estimator. Numerical results presented both for the multigrid approach and the domain decomposition method confirm the theoretical findings.

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arXiv2026

Speeding up an unsteady flow simulation by adaptive BDDC and Krylov subspace recycling

We deal with accelerating the solution of a sequence of large linear systems solved by preconditioned conjugate gradient method (PCG). The sequence originates from time-stepping within a simulation of an unsteady incompressible flow. We apply a pressure correction scheme and focus on the solution of the Poisson problem for the pressure corrector. Its scalable solution presents the main computational challenge in many applications. The right-hand side of the problem changes in each time step, while the system matrix is constant and symmetric positive definite. The acceleration techniques are studied on a representative problem of flow around a unit sphere. Our baseline approach is based on a parallel solution of each problem in the sequence by nonoverlapping domain decomposition method. The interface problem is solved by PCG with the three-level BDDC preconditioner. As a preliminary step, an appropriate stopping criterion for the PCG iterations is chosen. Next, two techniques for accelerating the solution are gradually added to the baseline approach. Deflation is used within PCG with several approaches to Krylov subspace recycling. Finally, we add the adaptive selection of the coarse space within the three-level BDDC method. The paper is rich in experiments with careful measurements of computational times on a parallel supercomputer. The combination of the acceleration techniques eventually leads to saving more than 40 % of the computational time.

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arXiv2024

A posteriori error estimates based on multilevel decompositions with large problems on the coarsest level

Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper deals with residual-based error estimates that are based on properties of quasi-interpolation operators, stable-splittings, or frames. We focus on the settings where the system matrix on the coarsest level is still large and the associated terms in the estimates can only be approximated. We show that the way in which the error term associated with the coarsest level is approximated is substantial. It can significantly affect both the efficiency (accuracy) of the overall error estimates and their robustness with respect to the size of the coarsest problem. The newly proposed approximation of the coarsest-level term is based on using the conjugate gradient method with an appropriate stopping criterion. We prove that the resulting estimates are efficient and robust with respect to the size of the coarsest-level problem. Numerical experiments illustrate the theoretical findings.

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arXiv2023

Estimating the error in CG-like algorithms for least-squares and least-norm problems

In [Meurant, Papež, Tichý; Numerical Algorithms 88, 2021], we presented an adaptive estimate for the energy norm of the error in the conjugate gradient (CG) method. In this paper, we extend the estimate to algorithms for solving linear approximation problems with a general, possibly rectangular matrix that are based on applying CG to a system with a positive (semi-)definite matrix build from the original matrix. We show that the resulting estimate preserves its key properties: it can be very cheaply evaluated, and it is numerically reliable in finite-precision arithmetic under some mild assumptions. We discuss algorithms based on Hestenes-Stiefel-like implementation (often called CGLS and CGNE in the literature) as well as on bidiagonalization (LSQR and CRAIG), and both unpreconditioned and preconditioned variants. The numerical experiments confirm the robustness and very satisfactory behaviour of the estimate.

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arXiv2021

Accurate error estimation in CG

In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations $Ax=b$ with a real symmetric positive definite matrix $A$. During the iterations it is important to monitor the quality of the approximate solution $x_k$ so that the process could be stopped whenever $x_k$ is accurate enough. One of the most relevant quantities for monitoring the quality of $x_k$ is the squared $A$-norm of the error vector $x-x_k$. This quantity cannot be easily evaluated, however, it can be estimated. Many of the existing estimation techniques are inspired by the view of CG as a procedure for approximating a certain Riemann--Stieltjes integral. The most natural technique is based on the Gauss quadrature approximation and provides a lower bound on the quantity of interest. The bound can be cheaply evaluated using terms that have to be computed anyway in the forthcoming CG iterations. If the squared $A$-norm of the error vector decreases rapidly, then the lower bound represents a tight estimate. In this paper we suggest a heuristic strategy aiming to answer the question of how many forthcoming CG iterations are needed to get an estimate with the prescribed accuracy. Numerical experiments demonstrate that the suggested strategy is efficient and robust.

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