Adaptivity in numerical PDEs refers to the data-dependent adaptation of the numerical discretisation of PDEs with the aim of improving the quality-cost balance. It is often only adaptivity that allows precise numerical simulation of complex physical processes. As the target function is typically unknown, adaptivity is usually organised iteratively, with decisions being made in each iteration based on a posteriori error indicators to adaptively improve the next iteration. Because of the early recognized enormous advantages of adaptive methods in practical applications it has become an indispensable tool in scientific computing and engineering sciences. In addition, a well-founded mathematical theory for basic modelling problems has been developed over the last three decades, which has been recently successively extended to larger problem classes, such as time-dependent problems, and includes more and more aspects, such as adaptive numerical solvers and strictly equivalent a posteriori bounds.
The workshop aims to present the latest developments in practical applications and theoretical findings in the field of adaptivity.
This workshop will have connections the the FoCM workshops “Approximation Theory and Computational Harmonic Analysis”, “Foundations of Numerical PDEs”, and “Numerical Linear Algebra”.
Organizers
Speakers
Invited speakers
Chemnitz University of Technology
University of Birmingham
TU Wien
RWTH Aachen
INRIA
Pontificia Universidad Católica de Chile
Friedrich Schiller University Jena
University of Twente
University of Bonn
University of Florence
Pontificia Universidad Católica de Chile
University of Limerick
Sorbonne University
TU Wien
Universidad Técnica Federico Santa María
TU Wien
Bielefeld University
Leipzig University
University College London
Leipzig University
TU Darmstadt
Università degli Studi di Milano
INRIA Paris
Università degli Studi di Milano
Thursday, 16 July
![[HS 14 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS14.png){kind=link}
14:00-14:30
Alex Bespalov (University of Birmingham)
We propose a multimesh rational approximation scheme for the numerical solution of the (homogeneous) Dirichlet problem for the spectral fractional Laplacian. Building on the works of Bonito and Pasciak [2] and Bulle et al [3], the scheme combines a rational approximation of a power function with a family of finite element discretizations of parameter-dependent, non-fractional partial differential equations (PDEs). The key novel idea that underpins the proposed multimesh scheme is that each parametric PDE is numerically solved on an individually tailored finite element mesh, see [1].
In this talk, we discuss an a posteriori error estimation strategy for the proposed rational approximation scheme and an adaptive multimesh refinement algorithm. We then present the results of numerical experiments that demonstrate that our adaptive multimesh approach yields significant reductions in computational costs when compared to an adaptive approach based on a single finite element mesh.
References
[1] A. Bespalov and R. Bulle, An adaptive multimesh rational approximation scheme for the spectral fractional Laplacian, SIAM J. Sci. Comput., (2026), to appear (arXiv:2504.03408).
[2] A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp., 84 (2015), pp. 2083-2110.
[3] R. Bulle, O. Barrera, S. P. A. Bordas, F. Chouly, and J. S. Hale, An a posteriori error estimator for the spectral fractional power of the Laplacian, Comput. Methods Appl. Mech. Engrg., 407 (2023), Paper No. 115943.
Joint work with Raphaël Bulle (Centre Inria de l’Université de Lorraine).
14:30-15:00
Dietmar Gallistl (Friedrich Schiller University Jena)
This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a re-start is introduced after a certain amount of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations.
15:00-15:30
Gregor Gantner (University of Twente)
We consider general linear second-order PDEs on unbounded domains and discretize them by adaptive Lagrange finite elements. To obtain finite-dimensional spaces, given an infinite mesh of the full domain, finite element approximations are computed on an active finite submesh, leading to an approximation of the solution by zero in the remainder of the active region. Under appropriate growth conditions of the full mesh, a standard weighted-residual error estimator can be shown to be reliable and efficient. We use this estimator to steer an adaptive algorithm that automatically refines the full mesh and arbitrarily enlarges the active submesh where necessary. We prove that this algorithm converges at optimal rates with respect to the number of active mesh elements. Moreover, we show that the unknown solution can be approximated at a certain rate if it can be approximated at that rate on a fixed bounded subdomain and satisfies certain smoothness and decay conditions on the unbounded complement of this subdomain. Finally, we provide numerical examples illustrating our key theoretical findings.
15:30-16:00
Carlotta Giannelli (University of Florence)
Adaptive isogeometric analysis (IGA) provides a powerful framework for the numerical solution of partial differential equations by combining CAD-based geometry representation with advanced spline discretizations. This talk presents recent developments in adaptive IGA, focusing on methodological advances, practical applications, and anisotropic spline refinement. We first discuss adaptive strategies for complex geometries, including multi-patch domains, where continuity constraints and localized refinement require careful treatment. We then consider applications to phase-field models, including the Cahn–Hilliard equation and coupled tumor growth models, demonstrating the effectiveness of adaptive IGA in capturing interface dynamics and multiscale phenomena in both two and three dimensions. Finally, we address hierarchical spline constructions underlying these adaptive schemes, with particular emphasis on recent developments enabling anisotropic local refinement. These approaches allow directional resolution of localized features while preserving key properties such as linear independence and local support.
16:00-16:30 Coffee Break
16:30-17:00
Dirk Praetorius (TU Wien)
Our talk addresses optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the setting of the Lax-Milgram theorem. We present recent work [arXiv:2604.17947] that formulates 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.
It is shown that the algorithmic interplay of mesh-refinement and iterative solver is optimal: A 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 convergence. Moreover, for sufficiently small adaptivity parameters, the quasi-error decays with optimal rates with respect to the overall computational complexity.
The talk is based on joint work with Thomas Führer (PUC Santiago), Ani Miraci (Sorbonne, Paris), and Paula Hilbert (TU Wien).
17:00-17:30
Ani Miraci (Sorbonne University)
In this talk, we present 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 and a norm-contractive algebraic solver. We primarily focus on parameter-less choices like the Kačanov linearization method (also called Picard iteration in the literature) combined with an optimal geometric multigrid method (or likewise an optimally preconditioned CG method) for the arising system of linear equations. Crucially, we design the adaptive algorithm based on a novel parameter-free algebraic stopping criterion which exploits the energy structure of the problem. This allows to prove as a first result that the adaptively steered number of algebraic solver steps is uniformly bounded. Next, we establish that the adaptive algorithm guarantees full R-linear convergence, meaning that essentially each algorithmic step (either mesh-refinement, linearization, or algebra iteration) contracts a quasi-error quantity consisting of total error and a-posteriori error estimator. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the main and novel result of this approach consists in proving that the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, unconditional 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. Numerical experiments confirm the theoretical results.
17:30-18:00
Philipp Bringmann (TU Wien)
The Zarantonello fixed-point iteration is an established linearization scheme for nonlinear PDEs. This talk presents a weighted least-squares minimization for the simultaneous computation of the primal and dual solution of the linearized PDE. The resulting formulation allows for a conforming finite element discretization of both variables with arbitrary polynomial degree. The least-squares functional provides a built-in a posteriori discretization error estimator in each linearization step motivating an adaptive Uzawa-type algorithm with an outer linearization loop and an inner adaptive mesh-refinement loop. For quasilinear PDEs in divergence form satisfying a 2-growth condition, this guarantees R-linear convergence of the computed linearization iterates regardless of the initial guess. Numerical experiments illustrate the performance of the proposed algorithm.
18:00-18:30
Wolfgang Dahmen (RWTH Aachen)
Generating nonlinear reduced models for parametric families of PDEs or learning associated parameter-to-solution maps often leads to minimizing a convex energy in a Hilbert space over a nonlinear hypothesis class. Hence, optimizing over the budget of trainable weights, defining the hypothesis class, becomes (highly) non-convex.
In this talk we discuss adaptive strategies for enhancing robustness and accuracy of optimization
success, revolving around “conditional convergence guarantees”. The main conceptual constituents are natural gradient flows in combination with an adaptive network expansion. Adaptation aims at increasing alignment with an ideal descent path determined by the gradient in the ambient (infinite dimensional) Hilbert space. This can be further combined with perturbed proximal point methods that can be understood as approximate implicit Euler schemes for the ideal Hilbert space gradient flow. The rationale is that each proximal point step is (much) easier to realize than the global optimization task because in each step the initial guess gets the closer to the optimum, the smaller the step-size. This adaptation to an implicit ideal Hilbert space gradient directions is of interest because the dynamical systems arising in the context of natural gradient flows are generically extremely stiff or even DAEs. Exploiting stable variational formulations of the PDE system allows one to derive in certain relevant cases a posteriori criteria to check whether an approximate proximal step is sufficiently accurate to guarantee convergence. Roughly, network expansions are triggered when these criteria fail to be met even when the step-size reaches some lower bound. In this sense the results can be understood as “conditional convergence” with accuracy certificates.
Part of the material draws on joint work with Zhu Wang and Wuchen Li, University of South Carolina
Friday, 17 July
14:00-14:30
Thomas Führer (Pontificia Universidad Católica de Chile)
In this talk I present some recent results where we show how to define higher-order polynomial approximations from lower-order piecewise polynomial approximations. Under certain assumptions, which are met for solutions of many finite element methods, we prove that the novel approximation converges at higher rates.
We present these ideas in the framework of quasi-interpolators. These are defined by taking a weighted average of the orthogonal projection onto piecewise polynomials.
We prove that the novel operators enjoy optimal approximation properties. This can be exploited to define postprocessed solutions of finite element methods such as (hybridizable) discontinuous Galerkin methods. While popular postprocessing techniques achieve the same order of convergence and can be computed more efficiently, our approach extends to more general situations where, e.g., discrete gradient approximations are not directly accessible. Our findings can be used to define reliable a posteriori error estimators.
Throughout the talk we present numerical examples.
Our main results are based on technical results that analyze the intersection of orthogonal polynomials on patches.
14:30-15:00
Julian Rolfes (Bielefeld University)
We study the regularity of discrete solutions to the Poisson problem. This is related to the properties of the Ritz projection. We present two different approaches via Greens functions and Campanato estimates.
15:00-15:30
Johannes Storn (Leipzig University)
This talk introduces randomized projection operators onto piecewise polynomial spaces based on sampling and discrete least-squares polynomial approximation. The resulting operators are computable for arbitrary functions in L^2 and achieve near-optimal approximation properties in both L^2 and H^-1. When used as smoothers for incomplete or rough data, they lead to computable finite element discretizations that attain optimal convergence rates without any additional hidden assumptions on the right-hand side beyond L^2 regularity.
15:30-16:00
Tabea Tscherpel (TU Darmstadt)
The L2-projection mapping to Lagrange finite element spaces is an important tool in numerical analysis. Its Sobolev stability is known to be key to discrete stability and quasi-optimality estimates for parabolic problems. For adaptively generated meshes the proof of Sobolev stability is challenging and requires conditions on how strongly the mesh size varies.
We present stability properties under certain conditions on the polynomial degree, on the space dimension and on the mesh grading. This includes results for adaptively refined simplicial meshes in general space dimension and on conforming 2D hybrid meshes additionally containing general convex quadrilaterals.
This is joint work with Lars Diening (Bielefeld University), Viktoria Lingert and Johannes Storn (Universität Leipzig).
16:00-16:30 Coffee Break
16:30-17:00
Andreas Veeser (Università degli Studi di Milano)
We consider a nonlinear problem that can be cast in a well-posed weak form, where the test space is a Banach space. We can then define a metric on the trial space that makes the underlying differential operator a Lipschitz isometry.
Next, we discretize the weak form by a conforming (or nonconforming) Petrov-Galerkin method and suppose that it is well-posed, in the sense that the underlying discrete operator is bijective. In the above setting, the Lipschitz constant of the discrete operator is smaller than 1, while the Lipschitz constant of its inverse is related to a straightforward generalization of the inf-sup constant to Lipschitz maps.
Our main result is a characterization of the Lipschitz inf-sup constant, covering its duality and the existence of a Fortin operator. This allows us to transfer the stability and quasi-optimality results of the linear case.
We illustrate the potential of this abstract theory by applying it to the p-Laplacian and the obstacle problem.
17:00-17:30
Zhaonan Dong (INRIA)
We devise and analyse a novel 𝐻(𝐜𝐮𝐫𝐥)-reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes. The (non-polynomial) reconstruction is devised over the mesh vertex patches using the partition of unity induced by hat basis functions in combination with local Helmholtz decompositions. Our main focus is on homogeneous tangential boundary conditions. We prove that the difference between the reconstructed 𝐻(𝐜𝐮𝐫𝐥)-field and the original, piecewise polynomial field, measured in the broken curl norm and in the $L^2$-norm, can be bounded in terms of suitable jump norms of the original field. The bounds are always $h$-optimal, and $p$-suboptimal by $\frac12$-order for the broken curl norm and by $\frac32$-order for the $L^2$-norm. An auxiliary result of independent interest is a novel broken-curl, divergence-preserving Poincaré inequality on vertex patches. Moreover, the $L^2$-norm estimate can be improved to $\frac12$-order suboptimality under a (reasonable) assumption on the uniform elliptic regularity pickup for a Poisson problem with Neumann conditions over the vertex patches.
We also discuss extensions of the 𝐻(𝐜𝐮𝐫𝐥)-reconstruction operator to the prescription of mixed boundary conditions, to agglomerated polytopal meshes, and to convex domains.
Finally, we showcase an important application of the 𝐻(𝐜𝐮𝐫𝐥)-reconstruction operator to the $hp$-a posteriori nonconforming error analysis of Maxwell’s equations. We focus on the (symmetric) interior penalty discontinuous Galerkin (dG) approximation of some simplified forms of Maxwell’s equations.
17:30-18:00
Michael Karkulik (Pontificia Universidad Católica de Chile)
The ubiquitous Scott-Zhang interpolation operator on a finite element space has three distinguishing properties: (i) it is uniformly bounded in a scale of norms, (ii) it is a local projection onto the finite element space, and (iii) it reproduces discrete boundary values. Such operators have been constructed for a variety of stationary problems. In this talk, we adress the problem of defining such an operator for space-time finite elements for parabolic problems. This is joint work with Thomas Fuehrer, Gregor Gantner, and Roberto González.
18:00-18:30
David Niederkofler (TU Wien)
Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with a Radau IIA method that combines the advantages of the Crank-Nicolson and implicit Euler schemes. We obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error. Together with a reduced basis method that leverages the Laplace transform for building tailored subspaces of reduced dimension, we obtain a very efficient method.
Saturday, 18 July
14:00-14:30
Iain Smears (University College London)
describe the Nash equilibria in models of dynamic population games.
In this talk, we analyse a posteriori error bounds for stabilized finite element discretizations of second-order mean field games.
A central ingredient in the analysis is the proof of equivalence between the error, in suitable norms, and an appropriate dual norm of the residual, which requires a careful analysis of the nonlinear coupling in the system, as well as the nonnegativity of the density variable. We thereby obtain reliable and efficient a posteriori error estimators for the problem for a class of stabilised FEM.
We also present some original results of independent interest on the interaction between the stabilisation in the scheme and the a posteriori error estimators.
14:30-15:00
Pietro Zanotti (Università degli Studi di Milano)
This contribution provides guidelines to derive a posteriori error estimators for quasi-optimal nonconforming finite element discretizations of symmetric elliptic problems. The resulting estimators are defined for all source terms that are admissible to the underlying weak formulations. More importantly, they are equivalent to the error in a strict sense. In particular, their data oscillation part is bounded by the error and, furthermore, can be designed to be bounded by classical data oscillations. The estimators are computable, except for the data oscillation part. Since even the computation of some bound of the oscillation part is not possible in general, it must be handled on a case-by-case basis. The abstract framework is illustrated by examples involving second- and fourth-order problems.
15:00-15:30
Joscha Gedicke (University of Bonn)
This talk presents a new error estimation framework for the symmetric interior penalty discontinuous Galerkin (SIPDG) discretization of the two-dimensional biharmonic problem. The central contribution is a novel decomposition of an error measure based on a generalized Hessian into two distinct components: one measuring the conformity and the other the nonconformity of the scheme. Leveraging this splitting, we design a new error estimator that is provably reliable and efficient for polynomial degrees p≥3, notably without the need for any DG stabilization terms. Furthermore, we show that this estimator can be bounded from above by the standard DG residual-based error estimator. Numerical experiments are provided to validate the theoretical predictions, demonstrating the efficiency of the proposed estimator for all polynomial degrees p≥2.
15:30-16:00
Mira Schedensack (Leipzig University)
A shear-locking free finite element discretization of the Reissner-Mindlin plate model is introduced. The rotation is discretized with piecewise polynomials of degree k+2 while the degree k is used for the displacement gradient. The method is closely related to the (generalized) Taylor-Hood pairing. Besides the a priori error analysis, this talk introduces an error estimator and proves its reliability and efficiency. Numerical experiments show the ability of a corresponding adaptive algorithm to balance the refinement of boundary layers and singularities.
16:00-16:30 Coffee Break
16:30-17:00
Martin Vohralik (INRIA Paris)
We investigate discrete Poincaré inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl) and H(div) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the shape regularity and cardinality of the underlying tetrahedral mesh, and the polynomial degree. One important focus is on meshes being local patches (stars) of tetrahedra from a larger tetrahedral mesh. We also review various equivalent results to the discrete Poincaré inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincaré maps), and existence of graph-stable commuting projections.
17:00-17:30
Enrique Otarola (Universidad Técnica Federico Santa María)
We analyze a bilinear optimal control problem for the Stokes-Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.
17:30-18:00
Fleurianne Bertrand (Chemnitz University of Technology)
We propose a least-squares patch reconstruction of fluxes for adaptive conforming primal finite element methods. In contrast to classical equilibrated estimators, where the reconstructed flux is obtained from local mixed problems, our reconstruction uses local symmetric positive definite least-squares problems. The resulting flux is quasi-equilibrated and yields patchwise indicators for adaptive refinement. A perturbed Prager-Synge identity shows that the loss of exact equilibration appears as a computable higher order term
18:00-18:30
Natalia Kopteva (University of Limerick)
There is a substantial literature on the a-posteriori error estimation of Discontinuous Galerkin (DG) methods. One attractive feature of such methods, as well as other nonconforming methods with discontinuous computed solutions, is the flexibility in the mesh design. At the same time, the proofs of a posteriori error bounds are typically considerably more technical for DG and other nonconforming methods than the corresponding conforming FEMs since such proofs typically require a certain reconstruction/recovery of a piecewise discontinuous numerical solution into the natural (conforming) function space of the exact PDE problem solution. It should also be noted that such reconstructions typically induce additional restrictions on the nonconforming mesh. With regard to the pointwise a-posteriori error estimation for DG methods, it appears to have been addressed only in [1], in the context of the Laplace equation, and in [2], in the context of singularly perturbed reaction-diffusion equations. The analysis in [1] required a global continuous reconstruction of the DG computed solution, which lad to an assumption that the nonconforming mesh should be derived by a systematic refinement of an initial conforming mesh and the number of hanging nodes per edge should be uniformly bounded (which also implied local mesh size equivalence for the the coarsest conforming refinement). By contrast, it was noted in [2] that since the error is estimated pointwise, only a local continuous reconstruction of the computed solution (in a $O(h_{\min})$ neighbourhood of the point at which the error is estimated) is needed. Additionally, a few novel approaches to constructing such continuous reconstructions, both local and global, were introduced under significantly milder restrictions on the nonconforming mesh. In particular, mesh elements with arbitrarily numbers of hanging nodes were allowed under minimum restrictions on the structure of the nonconforming mesh. In this talk, we shall take a further step in simplifying the analyses [1,2] by entirely dropping the requirement of either global [1] or local [2] continuous reconstructions of discontinuous computed solutions in the proof. This is achieved by certain additional bounds on the flux of the gradient of the Green’s function associated with the original elliptic differential operator. Hence, the analysis works under minimum assumptions on the mesh. In the final part of the talk, I shall also discuss ongoing joint work with Zhaonan Dong (INRIA, Paris) on the pointwise a-posteriori error estimation for another class of nonconforming methods, Hybrid High-Order (HHO) methods. [1] A. Demlow, E. Georgoulis, Pointwise a posteriori error control for discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 50 (2012), 2159–2181. [2] N. Kopteva, R. Rankin, Pointwise a posteriori error estimates for discontinuous Galerkin methods for singularly perturbed reaction-diffusion equations, SIAM J. Numer. Anal., 61 (2023), 1938-1961.
Posters
![[PZ14 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Urmila Bosisio
(Università degli Studi di Milano)
The poster concerns the study of instance optimal algorithms for the adaptive approximation of a function where the degrees of freedom (DOFs) are organized in trees. A key ingredient of such algorithms is the so-called marking strategy which, with the help of local quantities called marking indicators, decides where to introduce new DOFs. These marking indicators are based on local errors, which in applications are
subadditive or, more general, stable. I propose a new type of marking indicator and prove that it leads to instance optimality.
I intertwine ideas from both [Binev, DeVore, Fast computation in adaptive tree approximation] and [Binev, Tree approximation for h-p adaptivity] and propose a new type of marking indicator. In particular, I compare my result with the ones already obtained in the literature, showing that my result requires fewer assumptions. The new proof for the new indicators may be considered a step forward towards a further weakening of the
assumptions, in the sense that the errors are stable only on so-called conforming leaves.
This would allow for a whole host of new applications in a FEM setting.
This work is a joint work with professor Andreas Veeser.
Maximilian Brunner
(TU Wien)
This poster presents the inclusion of Newton’s method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). The proposed algorithm features an adaptive choice of the damping parameter in the Newton iteration for the discretized nonlinear problems on each level, ensuring both global linear and local quadratic convergence. In contrast to energy-based arguments in the literature, a novel approach in the analysis considers the discrete dual norm of the residual as a computable measure for the linearization error. As a consequence, we present the first convergence analysis with optimal convergence rates of an adaptive iteratively linearized FEM beyond energy-minimization problems.
Sônia Gomes
(Universidade Estadual de Campinas)
The goal of this presentation is to discuss the main aspects of some fully computable a posteriori error estimators for approximate variables obtained from two mixed finite element formulations: for the fluid flow in porous media [1] and for the stress tensor in elasticity problems [2]. These models are of saddle-point type and arise from associated minimization problems in which specific constraints are imposed, namely the divergence constraint on the flux and stress variables, as well as stress symmetry. In these settings, the variational formulations represent the corresponding optimality conditions, which are enforced through the introduction of additional variables – Lagrange multipliers – whose role is to weakly impose the constraints. The adopted error estimation methodology is inspired by the Prager–Synge identity, which provides a natural framework for deriving guaranteed upper bounds for flux or stress errors measured in their energy norms. Since the mixed methods already yield equilibrated globally H(div)-conforming flux and stress approximations, the only remaining requirement is a H1-conforming reconstruction of the piecewise discontinuous approximation computed for the potential (pressure or displacement). The efficiency and accuracy of the proposed estimators are validated through theoretical analysis and numerical experiments involving adaptive mesh refinements.
[1] Batistela, de Siqueira, Devloo, Gomes. A posteriori error estimator for a multiscale hybrid mixed method for Darcy’s flows. IJNME 123:6052-6078, 2022.
[2] de Siqueira, Fernandes, Gomes, Quinelato. A posteriori error estimation for adaptive weakly symmetric mixed finite elements of elasticity problems. Preprint 2026
Paula Hilbert
(TU Wien)
We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with $2 \le N$ linear goal functionals.
While a naive approach must solve N+1 linear finite element systems to compute approximations of the primal solution and N dual solutions in each step of an adaptive algorithm, the proposed algorithm requires only to solve two linear finite element systems. Moreover, all finite element solutions are computed with respect to the same discrete space, while the underlying triangulations are adapted to resolve all inherent singularities simultaneously. Unlike available results for multigoal-oriented adaptive finite element methods in the literature, we give the first thorough convergence analysis and prove that our algorithm guarantees, in the usual sense, even optimal convergence rates. Numerical experiments underline the derived theoretical results.
Marc Hovemann
(Friedrich Schiller University Jena)
On this poster we present a PDE solver based on quarklets with the optimal rate of convergence. For that purpose we deal with the B-spline quarklets introduced around 2015 by Dahlke, Keding and Raasch. The quarklets are modified Cohen-Daubechies-Feauveau spline wavelets that are enriched with polynomials. They are compactly supported and can be expressed in terms of explicit formulas. Moreover, they provide stable systems for several function spaces. Hence, they can be used to discretize certain elliptic PDEs, which then can be solved with classical iterative schemes like the damped Richardson iteration. In order to obtain a PDE solver with optimal convergence, we combine the Richardson iteration with a coarsening routine that provides the quarklet tree approximation of the current approximate solution. The subroutine of adaptive quarklet tree approximation can be seen as $hp$-method and is known to be near-best. This property carries over to the overall PDE solver and implies its optimality. Several numerical experiments and rigorous proofs for some test functions indicate that both quarklet tree approximation and the quarklet PDE solver have exponential rates of convergence.
Jing Li
(East China Normal University)
In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element space to approximate the phase field, and the nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants to approximate the velocity field and the pressure field, respectively. We establish the convergence of the adaptive method, i.e., the sequence of minimizers contains a subsequence that converges to a solution of the first-order optimality system, and the associated subsequence of discrete pressure fields also converges. The analysis relies crucially on a new discrete compactness result of nonconforming linear finite elements over a sequence of adaptively generated meshes. We present numerical results for several examples to illustrate the performance of the algorithm, including a comparison with the uniform refinement strategy.
Christoph Lietz
(TU Wien)
This poster presents the first rigorous convergence analysis of the smoothed adaptive finite element method (S-AFEM) proposed in [Mulita, Giani, Heltai: SIAM J. Sci. Comput. 43, 2021]. S-AFEM modifies the classical adaptive finite element method (AFEM) by performing accurate discrete solves only on periodically determined mesh levels, while the intermediate levels employ a fixed number of cheap smoothing iterations. Numerical experiments in that work showed that this strategy generates adapted meshes comparable to those of AFEM at substantially lower computational cost. We prove unconditional full R-linear convergence of a suitable quasi-error quantity and, for sufficiently small adaptivity parameters, optimal convergence rates with respect to the overall computational cost. The analysis requires only a mild uniform stability assumption on the employed smoother, satisfied by standard methods such as Richardson, Gauss-Seidel, conjugate gradient, and multigrid schemes. Our results apply to general second-order linear elliptic PDEs and show that S-AFEM retains all desired abstract convergence guarantees of AFEM while reducing the cumulative computational time. Numerical experiments validate the theory, analyze runtime performance, and underline the potential of S-AFEM for speed-up in AFEM computations.
Fabio Zoccolan
(EPFL)
Inspired by the work on PDEs [BDS2025], we develop adaptive low-rank time-integration procedures that dynamically adjust the rank of the computed surrogate to achieve a prescribed accuracy while maintaining computational efficiency. Unlike [BDS2025], the proposed method applies to general ordinary and stochastic differential equations and does not rely on iterative time-stepping schemes. Instead, it can be interpreted as an explicit Runge–Kutta method augmented with carefully placed recompression steps that enforce rank control while preserving the desired accuracy.
The method operates fully online and requires no prior knowledge of the solution structure. This makes it competitive with fixed-rank online approaches such as dynamical low-rank approximation, which may suffer from uncontrolled approximation errors. We establish convergence guarantees for the resulting approximations.
Moreover, we show that the ranks selected by the adaptive procedure remain comparable to the minimal ranks required to approximate the solution at the achieved accuracy. Numerical experiments corroborate the theoretical findings and demonstrate the efficiency and robustness of the proposed approach.
Joint work with: Matthieu Dolbeault (Brown University) and Polina Sachsenmaier (RWTH Aachen).
[BDS2025] Markus Bachmayr, Matthieu Dolbeault, and Polina Sachsenmaier. “Iterative thresholding low-rank time integration.” arXiv preprint arXiv:2507.15848 (2025).
[DSZ2026] Matthieu Dolbeault, Polina Sachsenmaier, and Fabio Zoccolan. “Analysis and rank control for step-truncation methods for ODEs and SDEs” in preparation (2026).