Quadrature error estimates on non-matching grids in a fictitious domain framework for fluid-structure interaction problems

TL;DR

We study a fictitious-domain fluid-structure interaction model using a distributed Lagrange multiplier to enforce kinematic coupling, where the coupling term couples two independent non-matching meshes. The work proves discrete well-posedness under inexact integration and derives quadrature-error estimates, showing that the $c_0$ coupling is stable as $h_\mathcal{B}\to0$, while the $c_1$ coupling requires $h_\mathcal{B}/h_\Omega\to0$ for optimal convergence. A Strang-type framework is used to connect quadrature errors to overall accuracy, and explicit rates like $O(h_\mathcal{B}^{3/2}|\log h_\mathcal{B}^{min}|)$ for $c_0$ and $O(h_\mathcal{B}^{1/2}|\log h_\mathcal{B}^{min}|+h_\mathcal{B}/h_\Omega)$ for gradients are established. Numerical tests in two dimensions confirm the predicted rates and demonstrate the practical impact of mesh ratios on accuracy, consistent with prior investigations in this framework.

Abstract

We consider a fictitious domain formulation for fluid-structure interaction problems based on a distributed Lagrange multiplier to couple the fluid and solid behaviors. How to deal with the coupling term is crucial since the construction of the associated finite element matrix requires the integration of functions defined over non-matching grids: the exact computation can be performed by intersecting the involved meshes, whereas an approximate coupling matrix can be evaluated on the original meshes by introducing a quadrature error. The purpose of this paper is twofold: we prove that the discrete problem is well-posed also when the coupling term is constructed in approximate way and we discuss quadrature error estimates over non-matching grids.

Figures (6)

• Figure 1: Geometric configuration of the problem. A Lagrangian point $\mathbf{s}$ of the solid reference domain $\mathcal{B}$ is mapped into the actual position of the body $\Omega^s$ through the map $\mathbf{X}$. The domain $\Omega$ acts like a container.
• Figure 2: On the left, mapping of a solid element into the fluid mesh. On the right, the yellow support of the fluid basis function associated with the starred node only partially matches the mapped solid triangle.
• Figure 3: Example of meshes used for the discretization of Problem \ref{['pro:stationary_test']}. From left to right: a right-oriented uniform mesh (fluid), a left-oriented uniform mesh (solid) and the geometric configuration of the problem (fluid mesh in orange, mapped solid mesh in dark blue).
• Figure 4: Decay of the quadrature error committed when the coupling term is assembled with the approximate procedure. The quadrature error is measured by computing the 1--norm of the difference $\mathsf{C}_f - \mathsf{C}_{f,h}$. The results agree with the theoretical estimates presented in Section \ref{['sec:quad_error']}. More precisely, the $\mathbf{L}^2(\mathcal{B})$ coupling (blue line) black converges with rate 2in Test 1 (${h_\mathcal{B}\rightarrow0}$, $h_\mathcal{B}/h_\Omega=1/2$). On the other hand, the error related to the $\mathbf{H}^1(\mathcal{B})$ coupling (orange line) decays only in Test 2 ($h_\mathcal{B}\rightarrow0$, $h_\mathcal{B}/h_\Omega=\frac{1}{2}h_\mathcal{B}^{1/3}\rightarrow0$) with rate $1/3$, as expected.
• Figure 5: Convergence history of $\mathbf{u},p,\mathbf{X},\boldsymbol\lambda$ in Test 1: comparison between exact and approximate assembly of the interface matrix. The results obtained with $\mathbf{c}=\mathbf{c}_0$ are collected in the left column, while the right column is related to $\mathbf{c}=\mathbf{c}_1$.

Theorems & Definitions (29)

• Proposition 1
• Proposition 2
• Theorem 1
• Lemma 1
• proof
• Theorem 2
• Definition 1
• Lemma 2
• proof
• Proposition 3