Discretization Commute Research Articles

A stabilized finite element method for inverse problems subject to the convection\u2013diffusion equation. II: convection-dominated regime

We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection–diffusion equations and extend our previous analysis in Burman et al. (Numer. Math. 144:451–477, 2020) to the convection-dominated regime. Slightly adjusting the stabilized finite element method proposed for dominant diffusion, we draw upon a local error analysis to obtain quasi-optimal convergence along the characteristics of the convective field through the data set. The weight function multiplying the discrete solution is taken to be Lipschitz continuous and a corresponding super approximation result (discrete commutator property) is proven. The effect of data perturbations is included in the analysis and we conclude the paper with some numerical experiments.

Learning Adaptive Regularization for Image Labeling Using Geometric Assignment

We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determines the regularization properties of the assignment flow. Using the symplectic partitioned Runge–Kutta method for numerical integration, it is shown that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A convenient property of our approach is that learning is based on exact inference. Carefully designed experiments demonstrate the performance of our approach, the expressiveness of the mathematical model as well as its limitations, from the viewpoint of statistical learning and optimal control.

A discrete commutator theory for the consistency and phase error analysis of semi-discrete [formula omitted] finite element approximations to the linear transport equation

A novel method, based on a discrete commutator, for the analysis of consistency error and phase relations for semi-discrete continuous finite element approximation of the one-way wave equation is presented. The technique generalizes to arbitrary dimension, accommodates the use of compatible quadratures, does not require the use of complex calculations, is applicable on non-uniform mesh geometries, and is especially useful when conventional Taylor series or Fourier approaches are intractable. Following the theory the analysis method is demonstrated for several test cases.

High order variational integrators in the optimal control of mechanical systems

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.

On the use of the notion of suitable weak solutions in CFD

AbstractThe notion of suitable weak solutions for the three‐dimensional incompressible Navier–Stokes equations together with some standard regularization techniques for constructing these solutions is reviewed. The novel result presented in this paper is that Faedo–Galerkin weak solutions to the Navier–Stokes equations are suitable provided they are constructed using finite‐dimensional spaces having a discrete commutator property and satisfying a proper inf–sup condition. Low‐order mixed finite element spaces appear to be acceptable for this purpose. Connections between the notion of suitable solutions and LES modeling are investigated. A proposal for a large eddy scale model based on the notion of suitable solutions is made and numerically illustrated. Copyright © 2008 John Wiley & Sons, Ltd.

Faedo–Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable

Faedo–Galerkin weak solutions of the three-dimensional Navier–Stokes equations supplemented with Dirichlet boundary conditions in bounded domains are suitable in the sense of Scheffer [V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Comm. Math. Phys. 55 (2) (1977) 97–112] provided they are constructed using finite-dimensional approximation spaces having a discrete commutator property and satisfying a proper inf-sup condition. Finite element and wavelet spaces appear to be acceptable for this purpose. This result extends that of [J.-L. Guermond, Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable, J. Math. Pures Appl. (9) 85 (3) (2006) 451–464] where periodic boundary conditions were assumed.

Discrete Routh reduction

This paper develops the theory of Abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with Abelian symmetry. The reduction of variational Runge–Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical J2 correction, as well as the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a non-trivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the non-canonical nature of the symplectic structure.

Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable

It is shown in this paper that Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable provided they are constructed using finite-dimensional spaces having a discrete commutator property and satisfying a proper inf–sup condition. Low order mixed finite element spaces appear to be acceptable for this purpose. This question was open since the notion of suitable solution was introduced.

Les solutions élements finis des équations de Navier–Stokes périodiques en dimension trois sont « appropriées »

It is shown that the limits of Faedo–Galerkin approximations of the Navier–Stokes equations in the three-dimensional torus are suitable weak solutions to the Navier–Stokes equations provided they are constructed using finite-dimensional spaces having a discrete commutator property and satisfying a proper inf–sup condition. Low order mixed finite element spaces appear to be acceptable for this purpose. This question was open since the notion of suitable solution was introduced. To cite this article: J.-L. Guermond, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The discrete commutator property of approximation spaces

We prove a super approximation result for a wide class of approximation spaces (including finite elements and wavelet type spaces). In particular such property, also referred at as the discrete commutator property holds, provided that a projector operator exists, who acts “locally”. Such property is used in getting estimates through localization techniques ([3], [4], [6], [5]).