An $hp$ Error Analysis of HDG for Linear Fluid-Structure Interaction

TL;DR

This work develops a velocity-stress based linear FSI model solved with a monolithic, hp-adaptive HDG discretization. A global stress tensor couples the Stokes fluid and linear elastodynamics solid, with Crank–Nicolson time stepping ensuring stability and accuracy. Theoretical results establish well-posedness, energy stability, and hp convergence for the semi-discrete scheme, with comprehensive convergence analysis for the fully discrete method. Numerical experiments in 2D and 3D confirm the predicted rates, verify robustness under nearly incompressible limits, and demonstrate practical viability for linear FSI with thick structures.

Abstract

A variational formulation based on velocity and stress is developed for linear fluid-structure interaction (FSI) problems. The well-posedness and energy stability of this formulation are established. To discretize the problem, a hybridizable discontinuous Galerkin method is employed. An $hp$-convergence analysis is performed for the resulting semi-discrete scheme. The temporal discretization is achieved via the Crank-Nicolson method, and the convergence properties of the fully discrete scheme are examined. Numerical experiments validate the theoretical results, confirming the effectiveness and accuracy of the proposed method.

Figures (6)

• Figure 1: Computed errors versus the polynomial degree $k$ with $h=1/8$ and $\Delta t = 10^{-4}$. The errors are measured at $t=T=0.3$, by employing the coefficients \ref{['L1']}. The exact solution is provided by \ref{['exactSol']}.
• Figure 2: Fluid and solid domains.
• Figure 3: Numerical solutions of Example 2 at final time $T = 0.012$s, obtained with the discretization parameters $h= 0.1$, $k=2$, $\Delta t = 10^{-4}$, and different incompressibility parameters $\lambda_f$. Top: $x$--component of $\frac{2}{3} \boldsymbol{u}_{f,h}$ along bottom line $\Gamma^{bot}_f$; Middle: pressure along bottom line $\Gamma^{bot}_f$; Bottom: $y$--component of $\boldsymbol{d}$ along the interface $\Sigma$.
• Figure 4: The computational domain partitioned with a mesh size $h = 0.25$.
• Figure 5: Numerical solutions of Example 3 at progressively increasing final times. Discretization parameters used: mesh size $h= 0.25$, polynomial degree $k=3$, and time step $\Delta t = 10^{-4}$. Top: Pressure distribution along the center line $\{(x,0,0)\ 0 < x < 5\}$. Bottom: $y$--component of the displacement on the line $\{(x, 0.55, 0) : \ 0 <x<5 \}$.

Theorems & Definitions (30)

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