Abstract. In this work, we propose and analyze a new hybrid high-order method for the Brinkman problem for fluids with power-law viscosity. The proposed method supports general meshes and arbitrary approximation orders and is robust in all regimes, from pure (power-law) Stokes to pure Darcy. Robustness is reflected by error estimates that distinguish the contributions from Stokes- and Darcy-dominated elements as identified by an appropriate dimensionless number and that additionally account for pre-asymptotic orders of convergence. Theoretical results are illustrated by a complete panel of numerical experiments.
Abstract. Recent years have seen growing interest in solving partial differential equations (PDEs) using nonlinear approximation spaces, where tunable parameters control the basis functions and allow the discretization to adapt dynamically to the solution. While these approaches often perform well in practice, convergence guarantees for the underlying optimization problem remain scarce. This work develops a general optimization framework for energy minimization problems arising from linear self-adjoint elliptic PDEs, formulated over nonlinear approximation spaces admitting a natural split between linear and nonlinear parameters. Linear variables are updated via linear solves or steepest descent, while nonlinear variables are handled using projected gradient descent. We establish local convergence in gradients and in values under modular structural assumptions, including differentiability, boundedness, regularity, and directional convexity, and provide explicit convergence rates in terms of the structural constants of the problem. These assumptions are verified for a broad class of nonlinear approximation spaces, including adaptive finite elements, free-knot splines, kernel methods, and weight-constrained neural networks. Numerical experiments illustrate the benefits of the parameter split and several key aspects of the framework. In a recent work [Magueresse, Badia (2025, arXiv:2508.17705)], we apply the framework to overlapping free-knot tensor-product B-splines, obtaining a geometrically adaptive solver with rigorous convergence guarantees.
Abstract. This paper studies the commonly utilized windowed Anderson acceleration (AA) algorithm for fixed-point methods, [Formula: see text]. It provides the first proof that when the operator [Formula: see text] is linear, symmetric, and contractive, the windowed AA, which uses a sliding window of prior iterates, improves the root-linear convergence factor over the fixed-point iterations. When [Formula: see text] is nonlinear, yet has a symmetric, non-scalar, and locally contractive Jacobian at a fixed point, a slightly modified AA algorithm is proved to have an analogous root-linear convergence factor improvement over fixed-point iterations. Simulations verify our observations. Furthermore, experiments with different data models demonstrate AA is significantly superior to the standard fixed-point methods for Tyler’s M-estimation.
Abstract. This paper develops and discusses a residual-based a posteriori error estimator for parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. Based on the derived error indicator, a space-time adaptive algorithm is proposed. Numerical experiments illustrate and complement the theory.
Abstract. In this paper we consider an evolving surface finite element method method for the advection and diffusion of a scalar quantity on a moving closed curve. The diffusion process is controlled by a forcing term that may include a rough term (specifically a stochastic noise) which in particular destroys the classical time differentiability properties of the solution. We provide a suitable variational solution concept and a fully discrete finite element method discretization. Our error analysis appropriately generalizes classical estimates to this weaker setting. We present some numerical simulations that confirm our theoretical findings.
