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A posteriori error analysis of a robust virtual element method for stress-assisted diffusion problems

Franco Dassi, Rekha Khot, Andres E. Rubiano, Ricardo Ruiz-Baier

TL;DR

This work develops residual-based a posteriori error estimators for a robust virtual element discretisation of a nonlinear stress-assisted diffusion problem that couples elasticity and diffusion through a perturbed saddle-point formulation. By exploiting a parameter-robust global inf-sup condition, Helmholtz decompositions on $\mathbf{H}(\mathrm{div},\Omega)$, and quasi-interpolation operators for Stokes and edge VE spaces, the authors derive reliable and efficient estimators that remain robust across parameter regimes in both 2D and 3D. The 2D analysis is complemented by a complete 3D extension, including new quasi-interpolators and a 3D Helmholtz framework, with numerical experiments showing optimal convergence and effective adaptivity on polygonal meshes. The results advance robust a posteriori error control for VEM in multi-physics diffusion problems and support adaptive strategies on general polytopal meshes for high-contrast/complex geometries.

Abstract

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem in two and three dimensions. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition and Helmholtz decomposition for $\mathbf{H}(\mathrm{div}, Ω)$ lead to a reliable and efficient error estimator based on appropriately weighted norms that ensure parameter robustness. The a posteriori error analysis uses quasi-interpolation operators for Stokes and edge virtual element spaces, and we include the proofs of such operators with estimates in 3D for completeness. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.

A posteriori error analysis of a robust virtual element method for stress-assisted diffusion problems

TL;DR

This work develops residual-based a posteriori error estimators for a robust virtual element discretisation of a nonlinear stress-assisted diffusion problem that couples elasticity and diffusion through a perturbed saddle-point formulation. By exploiting a parameter-robust global inf-sup condition, Helmholtz decompositions on , and quasi-interpolation operators for Stokes and edge VE spaces, the authors derive reliable and efficient estimators that remain robust across parameter regimes in both 2D and 3D. The 2D analysis is complemented by a complete 3D extension, including new quasi-interpolators and a 3D Helmholtz framework, with numerical experiments showing optimal convergence and effective adaptivity on polygonal meshes. The results advance robust a posteriori error control for VEM in multi-physics diffusion problems and support adaptive strategies on general polytopal meshes for high-contrast/complex geometries.

Abstract

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem in two and three dimensions. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition and Helmholtz decomposition for lead to a reliable and efficient error estimator based on appropriately weighted norms that ensure parameter robustness. The a posteriori error analysis uses quasi-interpolation operators for Stokes and edge virtual element spaces, and we include the proofs of such operators with estimates in 3D for completeness. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.

Paper Structure

This paper contains 32 sections, 26 theorems, 152 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Recurrent notation
  4. The stress-assisted diffusion problem
  5. Model problem
  6. Assumptions on the nonlinear terms
  7. Weak formulation
  8. Parameter-dependent norms and robust solvability
  9. Virtual element discretisation
  10. Mesh assumptions
  11. Polynomial spaces
  12. Discrete spaces
  13. Projection and interpolation operators
  14. The virtual element formulation for the stress-assisted diffusion problem
  15. A posteriori error analysis
  16. ...and 17 more sections

Key Result

Theorem 1

\newlabelwell-posedness0 Define the ball $\mathrm{W} =\left\{ w \in \mathrm{Q}_2 \colon \lVert w\rVert_{\mathrm{Q}_2} \leq C_2 \sqrt{M} \left(\lVert\varphi_{\mathrm{D}}\rVert_{\frac{1}{2},\Gamma_{\mathrm{D}}} + \lVert g\rVert_{0,\Omega}\right) \right\}$. Suppose that $1\leq \lambda$, $0<\mu$, $\t where the corresponding constants $C_1$ and $C_2$ do not depend on the physical parameters.

Figures (8)

  • Figure 1: Extension of $\Omega$ (left) to a convex domain $B$ (right), with mixed boundary conditions on $\Gamma_N$ and $\Gamma_D$ (centre).
  • Figure 1: An illustration of the distinct meshes used in the examples.
  • Figure 2: Example 1. Behaviour of the error $\overline{\textnormal{e}}_*$, estimator $\Theta$ (left), and effectivity index $\textnormal{eff}$ (right) under uniform refinement across various meshes, polynomial orders, and parameter selections.
  • Figure 3: Example 2. Behaviour of $\overline{\textnormal{e}}_*$ (left column), $\Theta$ (middle column), and $\textnormal{eff}$ (right column) under uniform and adaptive refinement on L-shape (top row) and Australia-shape (bottom row) domains for a variety polynomial orders.
  • Figure 4: Example 2. Snapshots of the interest variables on the L-shape (top) and Australia-shape (bottom) meshes are shown for polynomial degrees $k_1=2$ and $k_2=1$ after 19 refinement steps.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • Proof 1
  • Lemma 3
  • ...and 31 more