Numerical discretizations of a convective Brinkman-Forchheimer model under singular forcing

TL;DR

This work studies a convective Brinkman–Forchheimer model with singular forcing in 2D Lipschitz domains by formulating the problem in weighted spaces $\mathbf{H}_0^1(\omega,\Omega)\times L^2(\omega,\Omega)/\mathbb{R}$ with $\omega\in A_2$. A fixed-point framework coupled with a Brinkman linear operator yields existence and uniqueness under a small-data condition, and a finite element discretization using mini and Taylor–Hood pairs achieves a Céa-type quasi-best bound in the energy norm on convex domains. A residual-based a posteriori error estimator is developed via a Ritz projection and local residuals, with reliability and local efficiency results, enabling an adaptive procedure that attains optimal experimental convergence rates even in the presence of singular sources (Dirac deltas) and in non-convex geometries. Numerical experiments demonstrate robustness of the estimator across convex and non-convex domains, multiple Dirac sources, and varying weight exponents, validating the theoretical results and the practical efficacy of the adaptive method.

Abstract

In two-dimensional Lipschitz domains, we analyze a Brinkman--Darcy--Forchheimer problem on the weighted spaces $\mathbf{H}_0^1(ω,Ω) \times L^2(ω,Ω)/\mathbb{R}$, where $ω$ belongs to the Muckenhoupt class $A_2$. Under a suitable smallness assumption, we prove the existence and uniqueness of a solution. We propose a finite element method and obtain a quasi-best approximation result in the energy norm \emph{à la Céa} under the assumption that $Ω$ is convex. We also develop an a posteriori error estimator and study its reliability and efficiency properties. Finally, we develop an adaptive method that yields optimal experimental convergence rates for the numerical examples we perform.

Figures (5)

• Figure 1: The initial meshes used in the adaptive algorithm, Algorithm 2, when (A.1)$\Omega = (0,1)^2$, (A.2)$\Omega=(-1,1)^2 \setminus[0,1)\times (-1,0]$, and (A.3)$\Omega=((-1.5,1.5)\times(0,1))\cup((-0.5,0.5)\times(-2,1))$.
• Figure 2: Example 1: Computational rates of convergence for $\mathcal{E}_{\alpha}(\mathbf{u}_{\mathscr{T}},\mathsf{p}_{\mathscr{T}};\mathscr{T})$ considering $\alpha \in \{0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75\}$ (B.1) and the meshes obtained after 20 adaptive refinements for $\alpha=0.5$ (156 elements and 85 vertices) (B.2); $\alpha=1.0$ (192 elements and 105 vertices) (B.3); and $\alpha=1.5$ (304 elements and 167 vertices) (B.4).
• Figure 3: Example 2: Computational rates of convergence for $\mathcal{E}_{\alpha}(\mathbf{u}_{\mathscr{T}},\mathsf{p}_{\mathscr{T}};\mathscr{T})$ considering $\alpha \in \{0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75\}$ (B.1) and the meshes obtained after 40 adaptive refinements for $\alpha=0.5$ (534 elements and 280 vertices) (B.2); $\alpha=1.0$ (1917 elements and 994 vertices) (B.3); and $\alpha=1.5$ (2401 elements and 1247 vertices) (B.4).
• Figure 4: Example 3: T--shaped domain with Dirac delta source points located at $\mathbf{z}_{1}=(0,0.5)$ and $\mathbf{z}_{2}=(0,-1)$.
• Figure 5: Example 3: Computational rate of convergence for $\mathcal{D}_{1.0}(\mathbf{u}_{\mathscr{T}},\mathsf{p}_{\mathscr{T}};\mathscr{T})$ (C.1); the mesh obtained after 60 adaptive refinements (4378 elements and 2263 vertices) (C.2); streamlines for $|\mathbf{u}_{\mathscr{T}}|$ (C.3); elevation for $|\mathbf{u}_{\mathscr{T}}|$ (C.4); pressure contour (C.5); and elevation for $\mathsf{p}_{\mathscr{T}}$ (C.6).

Theorems & Definitions (29)

• Definition 1: restricted class $A_p(D)$
• Theorem 2: well-posedness of the Brinkman problem
• Proof 1
• Lemma 3: boundedness of the convective and Forchheimer terms
• Proof 2
• Proposition 4: $\mathcal{T}_1:\mathfrak{B}_\mathsf{A}\to \mathfrak{B}_\mathsf{A}$ is a contraction
• Proof 3
• Theorem 5: well-posedness for small data
• Proof 4
• Theorem 7: well-posedness for small data