An unfitted HDG method for a distributed optimal convection-diffusion control problem

Abstract: We analyze a high order unfitted hybridizable discontinuous Galerkin (HDG) method for an optimal control problem governed by a convection-diffusion equation posed in a domain with piecewise-wise $\mathcal{C}<sup>2$ boundary $\partial Ω$. The computational domain $Ω_h$ does not necessarily fit $Ω$ and the Transfer Path Method (TPM) is used to transfer the boundary data from $\partial Ω$ to $\partial Ω_h$ through segments of direction $\boldsymbol{m}$. Under closeness conditions between $\partial Ω_h$ and $\partial Ω$ and on the transfer vector $\boldsymbol{m}$, we prove optimal order of convergence in the $L<sup>2$-norm for all variables of the state and adjoint problems. We also show numerical examples to complement the theory.

• The paper introduces an unfitted HDG method that leverages the Transfer Path Method to impose high-order Dirichlet data on non-conforming meshes for distributed control problems.
• It establishes optimal L2-error convergence rates of order k+1 under quantified geometric and stabilization conditions, effectively managing domain-mesh discrepancies.
• Numerical experiments on curved and nonconvex domains confirm the method’s robustness and efficiency, highlighting its potential for large-scale PDE-constrained optimization.

Unfitted HDG Methods for Distributed Optimal Convection-Diffusion Control Problems

Problem Statement and Motivation

The paper addresses the formulation, analysis, and implementation of an unfitted, high-order hybridizable discontinuous Galerkin (HDG) method for distributed optimal control problems governed by convection-diffusion PDEs in curved domains. The domain $\Omega$ possesses a piecewise-$\mathcal{C}^2$ boundary, and the control variable enters as a distributed (additive) source term.

A primary obstacle in numerically solving PDE-constrained optimization problems in practical scenarios is the pervasive occurrence of non-polygonal/geometrically complex domains, which do not align with standard finite element meshes. Traditional fitted mesh approaches either significantly increase computational costs due to remeshing (especially for moving or shape-optimization problems) or induce dominant geometric errors under unfitted techniques. The Transfer Path Method (TPM) has emerged as a high-order approach for managing such geometric discrepancies by transferring Dirichlet boundary data to non-fitted computational boundaries; however, rigorous analytical convergence results for unfitted HDG methods applied to control problems were, until now, incomplete.

Description of the Unfitted HDG-TPM Framework

The paper extends the HDG method to curved domains via the TPM, which enables high-order accuracy without body-fitted meshes. The essential components are:

• Domain and Meshes: The physical domain $\Omega$ is approximated by a computational domain $\Omega_h$ with mesh $\mathcal{T}_h$, not necessarily boundary-fitted. The computational boundary $\Gamma_h$ is separated from the true boundary $\Gamma = \partial\Omega$ by an $O(h^{1+\delta})$ gap.
• Transfer Path Construction: The TPM constructs a bijection $\phi: \Gamma_h \to \Gamma$, defines path segments along directions $\boldsymbol{m}$, and enables the exact boundary data to be imposed (up to a high order) on $\mathcal{C}^2$0 via line integration of the solution gradient extrapolated from the computational subdomain.
• Control Problem: The distributed optimal control problem seeks to minimize, over $\mathcal{C}^2$1,

$\mathcal{C}^2$2

subject to the convection-diffusion equation with Dirichlet data.

• HDG Formulation: The PDEs are recast into first-order mixed systems, discretized via high-order HDG finite element spaces $\mathcal{C}^2$3. The correct imposition of discrete Dirichlet data leverages TPM-based polynomial gradient extrapolation.
• Boundary Closeness and Path Conditions: Optimal convergence requires conditions on $\mathcal{C}^2$4's proximity to $\mathcal{C}^2$5, and on the transfer direction $\mathcal{C}^2$6's alignment with $\mathcal{C}^2$7's normal vector $\mathcal{C}^2$8. The paper rigorously quantifies admissibility, including allowance for $\mathcal{C}^2$9 not being strictly normal provided tangential components remain controlled, thus generalizing previous restrictive geometric settings.

Analytical Contributions

Rigorous Error Analysis

• Well-Posedness: The paper establishes existence and uniqueness of the discrete HDG system under the specified stabilization and path assumptions.
• A Priori Error Estimates: It proves that all state and adjoint variables achieve optimal $\Omega$0-error convergence rates of order $\Omega$1 (with $\Omega$2 the polynomial degree), provided the mesh and transfer paths satisfy quantified closeness criteria.
• Generality in Path Choice: Prior work required transfer paths to be normal to $\Omega$3; this analysis derives explicit quantitative bounds on allowed deviation from the normal and on stabilization parameters to preserve convergence.
• Handling Domain–Mesh Mismatch: The analysis precisely tracks the error induced by integration along (potentially non-short) transferring paths and the extrapolation of discrete gradients beyond $\Omega$4.
• Sharp Regularity Requirements: The results are established for Lipschitz domains with piecewise-$\Omega$5 boundaries; convexity is sufficient for the required elliptic regularity estimates.

Technical Innovations

• Extension of HDG Projections to Curved, Unfitted Domains: The mechanism for handling boundary data on $\Omega$6 via the TPM, including explicit error estimates for polynomial gradient extrapolation and the effect of tangential transfer vector misalignment, is detailed and generalized.
• Discrete Duality Arguments: Control of primal variable errors uses auxiliary (dual) optimal control problems, taking full account of the geometric mismatch (including non-vanishing errors at the computational boundary).
• Comprehensive Treatment of Transferring Path Error Accumulation: The approach develops sharp estimates on the accumulation of extrapolation error along transferring segments with explicit constants depending on geometric shape regularity, path deviation, and mesh parameters.

Numerical Validation

Numerical experiments in two dimensions, for convex smooth and nonconvex (kidney-shaped) domains, confirm that despite significant mesh–domain geometric mismatches (including the absence of a guaranteed higher-order boundary approximation), empirically observed convergence rates match the $\Omega$7 theoretical order for all variables. The method achieves optimal convergence even when $\Omega$8 (i.e., when the computational boundary does not converge to the physical boundary faster than $\Omega$9), which is outside the circumstances where the theory rigorously guarantees this rate—a result highlighted as a practical strength and potential avenue for further research.

Implications and Prospects

This work considerably strengthens the theoretical and practical foundation of high-order unfitted HDG methods for distributed PDE control on curved domains. The main implications are:

• Mesh Generation Efficiency: By rigorously supporting the use of background triangulations and non-body-fitted meshes, the method facilitates fast, robust simulation pipelines, especially valuable for optimization under geometry change (e.g., moving domains or shape optimization).
• Sharp Error Control in Unfitted Contexts: The analysis enables practitioners to rigorously set discretization and stabilization parameters even with nontrivial computational boundary geometry, with direct consequences for robust solver design in engineering applications.
• Future Developments: The results indicate strong practical optimality even when theoretical assumptions are violated, suggesting directions for future theoretical work (e.g., extending rigorous results to settings with only $\Omega_h$0 boundary approximation). In particular, coupling HDG-TPM approaches with adaptive mesh refinement or for interface and multiphysics problems is a logical next objective.

Conclusion

The paper delivers a mathematically rigorous analysis and practical demonstration of a high-order unfitted HDG method combined with the TPM for distributed optimal convection-diffusion control problems in curved domains. Optimal convergence is established under general geometric configurations, with quantified admissibility of transfer path deviation from the normal. The method’s empirical robustness in non-ideal mesh/domain configurations signals its applicability for large-scale control and shape optimization tasks where remeshing is impractical. This work marks a significant advancement in high-order unfitted finite element analysis for PDE-constrained optimization, with implications for both theory and application in computational sciences (2604.00211).