arXiv subjects

Ani Miraçi

Publications and source records attributed to Ani Miraçi.

7 recordsLinked to original sources

arXiv2026

Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity

We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers the local mesh-refinement as well as the termination of a generalized minimal residual solver (GMRES) with optimal preconditioner to solve the arising non-symmetric finite element systems. Algorithmic interplay of mesh-refinement and iterative solver is shown to be optimal: A natural and fully computable quasi-error monitoring discretization error and algebraic solver error guarantees unconditional convergence for any choice of adaptivity parameters, i.e., the algorithm cannot fail to converge. This is ensured algorithmically via a novel adaptive feedback-control for the solver-termination parameter that monitors and ensures full R-linear convergence. Finally, the quasi-error even decays with optimal rates with respect to the overall computational complexity if the adaptivity parameters are chosen sufficiently small.

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arXiv2026

Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity

We consider adaptive finite element methods (AFEMs) with inexact algebraic solvers for second-order symmetric linear elliptic diffusion problems. Optimal complexity of AFEM, i.e., optimal convergence rates with respect to the overall computational cost, hinges on two requirements on the solver. First, each solver step is of linear cost with respect to the number of degrees of freedom. Second, each solver step guarantees uniform contraction of the solver error with respect to the PDE-related energy norm. Both properties must be ensured robustly with respect to the local mesh size h (i.e., h-robustness). While existing literature on geometric multigrid methods (MG) or symmetric additive Schwarz preconditioners for the preconditioned conjugate gradient method (PCG) that are appropriately adapted to adaptive mesh-refinement satisfy these requirements, this paper aims to consider more general solvers. Our main focus is on preconditioners stemming from contractive solvers which need not be symmetrized to be used with Krylov methods and which are not only h-robust but also p-robust, i.e., the contraction constant is independent of the polynomial degree p. In particular, we show that generalized PCG (GPCG) with an h- and p-robust contractive MG as a preconditioner satisfies the requirements for optimal-complexity AFEM and that it numerically outperforms AFEM using MG as a solver. While this is certainly known for (quasi-)uniform meshes, the main contribution of the present work is the rigorous analysis of the interplay of the solver with adaptive mesh-refinement. Numerical experiments underline the theoretical findings.

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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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arXiv2025

Unconditional full linear convergence and optimal complexity of adaptive iteratively linearized FEM for nonlinear PDEs

We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization scheme (e.g., the Kačanov method) and a norm-contractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameter-free algebraic stopping criterion is designed and we prove that it leads to a uniformly bounded number of algebraic solver steps. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, parameter-robust convergence is guaranteed. Moreover, for sufficiently small adaptivity parameters, the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and, hence, time.

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arXiv2024

Iterative solvers in adaptive FEM

This chapter provides an overview of state-of-the-art adaptive finite element methods (AFEMs) for the numerical solution of second-order elliptic partial differential equations (PDEs), where the primary focus is on the optimal interplay of local mesh refinement and iterative solution of the arising discrete systems. Particular emphasis is placed on the thorough description of the essential ingredients necessary to design adaptive algorithms of optimal complexity, i.e., algorithms that mathematically guarantee the optimal rate of convergence with respect to the overall computational cost and, hence, time. Crucially, adaptivity induces reliability of the computed numerical approximations by means of a-posteriori error control. This ensures that the error committed by the numerical scheme is bounded from above by computable quantities. The analysis of the adaptive algorithms is based on the study of appropriate quasi-error quantities that include and balance different components of the overall error. Importantly, the quasi-errors stemming from an adaptive algorithm with contractive iterative solver satisfy a centerpiece concept, namely, full R-linear convergence. This guarantees that the adaptive algorithm is essentially contracting this quasi-error at each step and it turns out to be the cornerstone for the optimal complexity of AFEM. The unified analysis of the adaptive algorithms is presented in the context of symmetric linear PDEs. Extensions to goal-oriented, non-symmetric, as well as non-linear PDEs are presented with suitable nested iterative solvers fitting into the general analytical framework of the linear symmetric case. Numerical experiments highlight the theoretical results and emphasize the practical relevance and gain of adaptivity with iterative solvers for numerical simulations with optimal complexity.

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arXiv2023

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

We consider a general nonsymmetric second-order linear elliptic PDE in the framework of the Lax-Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive mesh-refinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method (AISFEM) leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

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arXiv2023

$hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree $p \ge 1$ and the (local) mesh size $h$. We further prove that the built-in algebraic error estimator which comes with the solver is $hp$-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

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