Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov--Galerkin, and Monotone Mixed Methods

Muga, Ignacio; Zee, Kristoffer G. van der

doi:10.1137/20m1324338

Cited by 16 publications

(30 citation statements)

References 32 publications

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“…In some sense duality mappings are a suitable nonlinear replacement for the Riesz map. This becomes particularly evident in the context of residual minimization problems; see, for example, [26].…”

Section: Outline Of the Papermentioning

confidence: 97%

“…In the previous section we have heuristically constructed the linear functional ℓt hat allowed us to define a suitable test function. In [26,Section 2] the same idea is undertaken in a rather more abstract setting. The functional ℓ˜can also be defined…”

Section: Duality Mappingsmentioning

confidence: 99%

“…For the purposes of this paper, there is no benefit to using this more general notion of duality mappings. However, for some practical aspects of using the method introduced in[26], a duality mapping with a different so-called weight is useful. An example of such a duality mapping is the functional ℓ .…”

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confidence: 99%

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The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

Houston

Muga

Roggendorf

et al. 2019

Computational Methods in Applied Mathematics

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While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space {H_{0}^{1}(\Omega)}, the Banach Sobolev space {W^{1,q}_{0}(\Omega)}, {1<q<{\infty}}, is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the {W^{1,q}_{0}(\Omega)}-{W_{0}^{1,q^{\prime}}(\Omega)} functional setting, {\frac{1}{q}+\frac{1}{q^{\prime}}=1}. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of {W^{1,q^{\prime}}}-stability of the {H_{0}^{1}}-projector.

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Section: Outline Of the Papermentioning

confidence: 97%

Section: Duality Mappingsmentioning

confidence: 99%

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The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

Houston

Muga

Roggendorf

et al. 2019

Computational Methods in Applied Mathematics

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“…More recently, a novel approach to designing finite element methods in a very general Banach space setting has been introduced in [17]. This approach is rooted in the so-called Discontinuous Petrov-Galerkin (DPG) methods [18] and extends the concept of optimal test norms and functions from Hilbert spaces to more general Banach spaces yielding a scheme that can be interpreted as a non-linear Petrov-Galerkin method.…”

Section: Introductionmentioning

confidence: 99%

Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation

Houston

Roggendorf

Zee

2020

Computers & Mathematics with Applications

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In this article we consider the numerical approximation of the convectiondiffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convectiondominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in L q -type Sobolev spaces, with 1 < q < ∞. We then apply a non-standard, non-linear Petrov-Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the convectiondiffusion-reaction equation, this yields a generalization of a similar approach from the L 2 -setting to the L q -setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples.

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“…The Petrov-Galerkin method is equivalent to a Minimal-Residual formulation, as commonly studied in the context of DPG and optimal Petrov-Galerkin methods [3,4]. As is natural in deep learning, we use an artificial neural network to define the family of test spaces, whose parameters are learned from the data.…”

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confidence: 99%

Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework

Zee¹,

Brevis²,

Muga³

2021

10th International Conference on Adaptative Modeling and Simulation

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We consider the data-driven acceleration of Galerkin-based finite element discretizations for the approximation of partial differential equations (PDEs) [1]. The aim is to obtain approximations on meshes that are very coarse, but nevertheless resolve quantities of interest with striking accuracy.Our work is inspired by the the machine learning framework of Mishra [2], who considered the data-driven acceleration of finite-difference schemes. The essential idea is to optimize a numerical method for a given coarse mesh, by minimizing a loss function consisting of errors with respect to the quantities of interest for obtained training data.

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Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov--Galerkin, and Monotone Mixed Methods

Cited by 16 publications

References 32 publications

The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation

Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework

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