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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

  • Paul Houston , Ignacio Muga ORCID logo , Sarah Roggendorf ORCID logo EMAIL logo and Kristoffer G. van der Zee ORCID logo EMAIL logo
Published/Copyright: June 27, 2019
Computational Methods in Applied Mathematics
From the journal Computational Methods in Applied Mathematics Volume 19 Issue 3

Abstract

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H01⁢(Ω), the Banach Sobolev space W01,q⁢(Ω), 1<q<∞, is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W01,q⁢(Ω)-W01,q′⁢(Ω) functional setting, 1q+1q′=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 W1,q′-stability of the H01-projector.

Keywords: Convection-Diffusion Equation; Inf-Sup Condition; Galerkin Methods; FEM; Well-Posedness; Banach Spaces; Elliptic Regularity
MSC 2010: 35A01; 35A02; 35J20; 65N12; 65N30

Funding source: Fondo Nacional de Desarrollo Científico y Tecnológico

Award Identifier / Grant number: 1160774

Funding source: FP7 People: Marie-Curie Actions

Award Identifier / Grant number: 777778

Funding statement: The work by Ignacio Muga was done in the framework of Chilean FONDECYT research project #1160774. Ignacio Muga was also partially supported by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 777778.

A Proof of Proposition 2.1

In this section, we give the proof of Proposition 2.1. We establish the inf-sup conditions (2.4a) and (2.4b), employing the standard duality technique that invokes the assumed regularity (i.e., Assumption 1.1). In the one-dimensional case, the latter assumption is not needed; see below. The proof is brief, since we employ straightforward properties of duality maps.

To proof (2.4a), let JW01,q denote the duality map of W01,q⁢(Ω) endowed with the (semi)norm |⋅|W1,q⁢(Ω) (see (2.8) for the expression of JW01,q). Take g=JW01,q⁢(u)∈W-1,q′⁢(Ω) and let zg solve (1.5) with r=q′>2. By Assumption 1.1, we get that zg∈W01,q′⁢(Ω) and

|zg|W1,q′⁢(Ω)=∥∇⁡zg∥Lq′⁢(Ω)≤Cq′,Ω⁢∥JW01,q⁢(u)∥W-1,q′=Cq′,Ω⁢|u|W1,q⁢(Ω).

Furthermore, by density, zg satisfies

∫Ω∇⁡w⋅∇⁡zg⁢d⁢𝒙=〈JW01,q⁢(u),w〉W-1,q′,W01,q for all ⁢w∈W01,q⁢(Ω)⊃H01⁢(Ω).

Next,

(A.1)

supv∈W01,q′⁢(Ω)⁡∫Ω∇⁡u⋅∇⁡v⁢d⁢𝒙|v|W1,q′⁢(Ω)≥∫Ω∇⁡u⋅∇⁡zg⁢d⁢𝒙|zg|W1,q′⁢(Ω)
=〈JW01,q⁢(u),u〉W-1,q′,W01,q|zg|W1,q′⁢(Ω)
=|u|W1,q⁢(Ω)2|zg|W1,q′⁢(Ω)
≥Cq′,Ω-1⁢|u|W1,q⁢(Ω),

which implies condition (2.4a) with γ=Cq′,Ω-1.

In order to prove condition (2.4b), let v∈W01,q′⁢(Ω) satisfy

∫Ω∇⁡w⋅∇⁡v⁢d⁢𝒙=0 for all ⁢w∈W01,q⁢(Ω).

In particular, for w=v∈W01,q′⁢(Ω)⊂W01,q⁢(Ω) we get

∥∇⁡v∥L2⁢(Ω)2=0,

which implies v=0. The a priori estimate (2.2) follows as usual from the inf-sup condition (A.1).

On the other hand, in the one-dimensional case (d=1), one can prove well-posedness without invoking Assumption 1.1. Indeed, let Ω⊂ℝ be any open bounded set. Since open sets are composed of a countable union of disjoint open intervals, we focus on the case of only one open interval, say for simplicity I=(0,1).

Let ρ>1 and notice that the image of the derivative operator (⋅)′ applied to W01,ρ⁢(I) gives the space (see [12, Lemma B.69])

L∫=0ρ⁢(I):={ϕ∈Lρ⁢(I):∫Iϕ⁢dx=0}.

Let σ=ρρ-1, and for any w∈W01,ρ⁢(I), define

ϕw=Jρ⁢(w′)-∫ΩJρ⁢(w′)⁢dx

using the duality map Jρ:Lρ⁢(I)→Lσ⁢(I) (see (2.7)). Observe that ϕw∈L∫=0σ⁢(I), w′∈L∫=0ρ⁢(I), and

(A.2)∥ϕw∥Lσ⁢(Ω)≤2⁢∥Jρ⁢(w′)∥Lσ⁢(Ω)=2⁢∥w′∥Lρ⁢(Ω).

Thus,

sup0≠v∈W01,σ⁢(Ω)⁡∫Ωw′⁢v′⁢dx∥v′∥Lσ⁢(Ω)=sup0≠ϕ∈L∫=0σ⁢(I)⁡∫Ωw′⁢ϕ⁢dx∥ϕ∥Lσ⁢(Ω) ⁢(by surjectivity)
≥∫Ωw′⁢ϕw⁢dx∥ϕw∥Lσ⁢(Ω) ⁢(since ϕw∈L∫=0σ⁢(I))
≥∫Ωw′⁢Jρ⁢(w′)⁢dx-∫ΩJρ⁢(w′)⁢dx⁢∫Ωw′⁢dx2⁢∥w′∥Lρ⁢(Ω) ⁢(by (A.2))
=12⁢∥w′∥Lρ⁢(Ω) ⁢(since w′∈L∫=0ρ⁢(I)).

In conclusion,

(A.3)sup0≠v∈W01,σ⁢(Ω)⁡∫Ωw′⁢v′⁢dx∥v′∥Lσ⁢(Ω)≥12⁢∥w′∥Lρ⁢(Ω)  for all ⁢ρ>1.

In particular, taking ρ=q, (A.3) implies the inf-sup condition (2.4a) for problem (2.1) with γ=12. On the other hand, taking ρ=q′, (A.3) implies that inf-sup condition (2.4b) is fulfilled.

Remark A.1 (Elliptic Regularity in One Dimension).

The proof of Proposition 2.1 shows that when d=1, problem (2.1) is well-posed in a W01,ρ-W01,σ-setting for anyρ>1, without any elliptic-regularity assumption. Of course, in particular this implies that when d=1, the elliptic-regularity Assumption 1.1 holds for any r>2.

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Received: 2018-08-05
Revised: 2019-03-29
Accepted: 2019-04-30
Published Online: 2019-06-27
Published in Print: 2019-07-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Special Issue Articles on Minimum Residual and Least-Squares Finite Element Methods
  3. Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods
  4. A Non-conforming Saddle Point Least Squares Approach for an Elliptic Interface Problem
  5. Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry
  6. Adaptive Strategies for Transport Equations
  7. Space-Time Discontinuous Petrov–Galerkin Methods for Linear Wave Equations in Heterogeneous Media
  8. Superconvergent DPG Methods for Second-Order Elliptic Problems
  9. The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method
  10. Fast Integration of DPG Matrices Based on Sum Factorization for all the Energy Spaces
  11. The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces
  12. Camellia: A Rapid Development Framework for Finite Element Solvers
  13. Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems
  14. A Newton Div-Curl Least-Squares Finite Element Method for the Elliptic Monge–Ampère Equation
  15. The Discrete Steklov–Poincaré Operator Using Algebraic Dual Polynomials
  16. Regular Research Articles
  17. A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction
  18. A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems
https://doi.org/10.1515/cmam-2018-0198
Keywords for this article
Convection-Diffusion Equation; Inf-Sup Condition; Galerkin Methods; FEM; Well-Posedness; Banach Spaces; Elliptic Regularity

Articles in the same Issue

  1. Frontmatter
  2. Special Issue Articles on Minimum Residual and Least-Squares Finite Element Methods
  3. Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods
  4. A Non-conforming Saddle Point Least Squares Approach for an Elliptic Interface Problem
  5. Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry
  6. Adaptive Strategies for Transport Equations
  7. Space-Time Discontinuous Petrov–Galerkin Methods for Linear Wave Equations in Heterogeneous Media
  8. Superconvergent DPG Methods for Second-Order Elliptic Problems
  9. The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method
  10. Fast Integration of DPG Matrices Based on Sum Factorization for all the Energy Spaces
  11. The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces
  12. Camellia: A Rapid Development Framework for Finite Element Solvers
  13. Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems
  14. A Newton Div-Curl Least-Squares Finite Element Method for the Elliptic Monge–Ampère Equation
  15. The Discrete Steklov–Poincaré Operator Using Algebraic Dual Polynomials
  16. Regular Research Articles
  17. A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction
  18. A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems
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