A Discontinuous Galerkin Method for Biharmonic Dirichlet Control Problems With Patch Reconstruction

1. Introduction

• Higher-order optimal control for biharmonic equations has received less attention than second-order control, and Dirichlet control is harder to analyze because it does not enter the variational formulation naturally.Prior approaches include very weak solutions for L2 boundary controls and penalization in H1/2(Γ), H1(Ω), or H2(Ω).Source: §1 · pp. 1–2
• The control acts on the normal derivative, whereas cited biharmonic Dirichlet-control methods impose it on the Dirichlet trace.The paper derives energy-norm estimates on convex polygons and L2 estimates when all interior angles are below 126.283 degrees.Source: §1 · p. 2
• Tangential Stokes boundary control motivates the biharmonic setting, with a stream function making the discrete velocity divergence-free automatically.The paper links the normal-derivative control to a stream-function formulation; the divergence-free property contrasts with a special scheme in cited work.Source: §1 · p. 2
• RDA provides a high-order approximation space with one unknown per element, reducing degrees of freedom to the mesh element count.The space is linearly isomorphic to piecewise constants and has optimal approximation properties under certain conditions.Source: §1 · p. 2

2. Optimal Control Problems

• The model assumes a convex polygonal domain and imposes zero state trace with the control as the normal derivative; the control space and regularization parameter are specified separately.The state depends uniquely on the control and forcing, making the cost functional quadratic in the control for the linear governing equation.Source: §2 · pp. 3–4
• The analysis treats L2(Γ) and H1(Γ) controls in a unified framework, while leaving the H1/2(Γ) energy-space choice for future work.For L2 controls the state is defined by transposition, whereas H1 controls admit an H2 state lifting.Source: §2 · p. 3
• Separating the state equation into forcing and control components yields the first-order optimality system for the state, control, and adjoint.The control-to-state map is linear and bounded, and its directional derivative is the zero-forcing state corresponding to that direction.Source: §2 · p. 4
• For a bounded convex polygon, the solution has the stated regularity, with additional regularity when every interior angle is below 126.283 degrees.The angle restriction permits the biharmonic solution operator to map L2(Ω) continuously into H4(Ω)∩H2_0(Ω).Source: §2 · pp. 4–5
• A smooth vector field with unit normal component on each open edge constructs a corner-compatible lifting, with its normal derivative recovering the Laplacian trace.The lifting satisfies ∂nGq = γ0(∆q) and is bounded from H^(k+1)(Ω) to H^k(Ω) for k=2,3.Source: §2 · p. 5
• Optimal-solution regularity depends on the control space: L2 controls yield a state in H2, while H1 controls gain H2 edgewise regularity.Under the angle restriction, the L2 case gives a state in H3; the H1 case gives an H4 adjoint and H2 control on each edge.Source: §2 · p. 5

3.1. The Reconstruction Operator.

• The operator reconstructs degree-m polynomials from one piecewise-constant value per element by patchwise constrained least squares, and is a linear isomorphism onto its image space.Patch size is chosen to provide enough sampling points; uniqueness holds under the stated assumptions.Source: §3.1 · p. 6
• For 2 ≤ s ≤ m + 1, reconstruction satisfies ∥z − Rm(Ihz)∥Hk(K) ≤ C h^(s−k)∥z∥Hs(S(K)) for 0 ≤ k ≤ s, recovering rate h^(m+1−k) when z ∈ Hm+1(Ω).The uniform estimate relies on patch approximation, overlap, and sampling assumptions; the patch cardinality rule alone does not ensure unisolvency.Source: §3.1 · p. 6

3.2. The Reconstructed Discontinuous Approximation Method.

• Because low-regularity solutions may lack traces required for Galerkin orthogonality, the analysis instead uses residual estimates and an enriching operator into a conforming HCT subspace.The stated example is ψ ∈ H^(3−δ)(Ω), 0 ≤ δ < 1/2, which generally lacks an L2 trace of ∇∆ψ.Source: §3.2 · p. 8
• If ψ ∈ H^(m+1)(Ω) and Osc(F) ≤ C h^(m−1)||ψ||_(H^(m+1)(Ω)), then the energy error is ≤ C h^(m−1)||ψ||_(H^(m+1)(Ω)); when every interior angle of Ω is less than 126.283 degrees, the L2 error is ≤ C h^(m+ϵ−1)||ψ||_(H^(m+1)(Ω)).Here ϵ = min{m−1, 2}.Source: §3.2 · p. 10
• For m ≥ 3, the approximation has order h^(m−1) when t = m + 1 and order h^2 when t = 4.Theorem 3 assumes a sufficiently large penalty; the order h^2 result holds for every m ≥ 3.Source: §3.2 · p. 11

3.3. The Discrete First Order Optimality System.

• Boundary controls use degree-r piecewise polynomials on each edge; H1 controls are continuous and corner-compatible, while L2 controls need continuity only within open macro-edges.The boundary mesh includes all polygonal vertices; L2 controls require no continuity across corners.Source: §3.3 · pp. 11–12
• The discrete scheme is defined directly from the optimality system, and equation (3.20) is the first-order optimality system of a discrete optimal control problem.The text says the scheme does not generally rely on equation (3.19).Source: §3.3 · p. 12

4. A Priori Error Estimates

• Boundary control projections use L2-orthogonal or H1 Ritz projections, with edgewise duality sharpening their L2(Γ) error bound to Ch times the H1(Γ) error.The duality argument solves edgewise problems with zero endpoint values.Source: §4 · p. 13

4.1. Estimates under high regularity.

• Under ψ,η ∈ H^(m+1) regularity, state and adjoint DG errors are O(h^(m−1)); the state L2 error is also O(h^(m−1)) without an angle restriction.These high-regularity estimates require additional smoothness assumptions, not guaranteed for arbitrary-degree problems on polygonal domains.Source: §4.1 · pp. 14–15

4.2. Estimates under the available polygonal regularity.

• With all interior angles below 126.283 degrees, the state energy-error rate is O(h) for U = L2(Γ) and O(h^(1−δ)) for U = H1(Γ).The low-regularity proof establishes both cases without a strong DG norm or Galerkin orthogonality for the state.Source: §4.2 · pp. 15–16

Example 1

• In Example 1, energy-norm errors converge close to order m−1, while H1 and L2 errors exceed m and approach m+1; control errors converge approximately at order m−1.These observed rates are generally sharper than the theoretical estimates.Source: §5 · p. 18

Example 2

• In Example 2, energy-norm convergence is at least order m−1, H1 and L2 convergence is near order m, and control convergence is approximately first order for m = 2 and generally order m for m = 3, 4.The reported rates are better than the theoretical estimates.Source: §5 · p. 19

Example 3

• In Example 3, state and adjoint errors are consistent with theory for sufficiently small h; control errors converge at order m for m = 2, 3 but near order 3 for m = 4.The authors suggest the m = 4 control rate may reflect the domain’s largest interior angle exceeding 126.283 degrees.Source: §5 · p. 21

Numerical scope

• The experiments use exact manufactured solutions, while testing limited-regularity cases is deferred because exact solutions and standard error comparisons are difficult with the non-nested RDA spaces.The paper identifies this as future work.Source: §5 · p. 21

• The method treats biharmonic Dirichlet boundary control with the control imposed on the normal derivative.Source: §6 · p. 21
• The discrete problem is well-posed, with a priori error estimates derived in the energy and L2 norms.Source: §6 · p. 21
• Numerical results substantiate the theoretical estimates and demonstrate the method’s practical effectiveness.Source: §6 · p. 21