Convergence and Optimality of Adaptive Mixed Methods on Surfaces

Michael Holst; Adam Mihalik; Ryan Szypowski

Home/Archive/arXiv

arXiv · 1404.1956

Convergence and Optimality of Adaptive Mixed Methods on Surfaces

Michael Holst·Adam Mihalik·Ryan Szypowski

View original source ↗Read source document ↗Export metadata ↓

Abstract

In a 1988 article, Dziuk introduced a nodal finite element method for the Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear triangulation, initiating a line of research into surface finite element methods (SFEM). Demlow and Dziuk built on the original results, introducing an adaptive method for problems on 2-surfaces, and Demlow later extended the a priori theory to 3-surfaces and higher order elements. In a separate line of research, the Finite Element Exterior Calculus (FEEC) framework has been developed over the last decade by Arnold, Falk and Winther and others as a way to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can be obtained by solving finite dimensional subproblems. In 2011, Holst and Stern merged these two lines of research by developing a framework for variational crimes in abstract Hilbert complexes, allowing for application of the FEEC framework to problems that violate the subcomplex assumption of Arnold, Falk and Winther. When applied to Euclidean hypersurfaces, this new framework recovers the original a priori results and extends the theory to problems posed on surfaces of arbitrary dimensions. In yet another seemingly distinct line of research, Holst, Mihalik and Szypowski developed a convergence theory for a specific class of adaptive problems in the FEEC framework. Here, we bring these ideas together, showing convergence and optimality of an adaptive finite element method for the mixed formulation of the Hodge Laplacian on hypersurfaces.

Explore related subjects

math.NA cs.NA

Keep this discovery

BibTeX RIS

Michael Holst, Adam Mihalik, Ryan Szypowski. 2014-04-07. Convergence and Optimality of Adaptive Mixed Methods on Surfaces. https://arxiv.org/abs/1404.1956

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity

In the Navier--Stokes equations, incompressibility allows rewriting the viscous term in various forms leading to distinct numerical properties and flow descriptions. Furthermore, models accounting for non-Newtonian, thermal or turbulent effects often break the constant-viscosity assumption, thereby producing additional consistency terms. In this context, the present work compares the classical symmetric-gradient diffusion term with more recent variable-viscosity generalizations of the Laplacian and rotational forms. We discuss, analyze and test their differences with respect to implementation, efficiency, numerical stability and outflow boundary conditions. With a focus on time-dependent flows, we consider second-order implicit-explicit (IMEX) temporal discretizations aimed at improving efficiency and numerical stability. Through a rigorous stability analysis, we show how selected explicit treatments can bypass algorithmic nonlinearities without inducing CFL conditions. Our numerical results highlight important differences between the three viscous formulations---especially in the presence of outflow boundaries, for which the generalized Laplacian form proves more suitable in diffusion-dominated regimes. %(as widely known for constant viscosity).

math.NA↗

arXiv2026

Full-window branch discovery and loss-selected EnKF continuation for data assimilation

We develop a framework for offline full-window branch discovery, optionally followed by online continuation with an ensemble Kalman filter (EnKF). Three mechanisms drive the branch search: adjoint path-kernel (APK) differentiation balances kernel differentiation and correction-stabilized path perturbation, shifting the optimization from exploration to exploitation; an optimized Gaussian initial law broadens the search over initial-state basins; and loss-weighted mixing across independent runs recombines successful path components. We may then select an interior state using a local loss and continue online with an EnKF. In 40-dimensional Lorenz-96 experiments, the mean offline path RMSE of APK is 4.3 times smaller than that of population weak-$\mathrm{4D\text{-}Var}_x$. The resulting APK-EnKF method has a mean online RMSE 64 times smaller than that of ordinary EnKF.

math.NA↗

arXiv2026

A variational physics-informed graph neural network for heterogeneous solid mechanics

Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])$, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the ($\sigma_{xx}$) error of an energy-based PINN ($5.01\%$ versus $10.94\%$). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.

math.NA↗