Local discontinuous Galerkin FEM for convex minimization

1. Problem Statement

The paper addresses the numerical approximation of convex minimization problems of the $p$-Laplace type. The continuous problem seeks to minimize an energy functional $E(v)$ over a Sobolev space $V = W^{1,p}_D(\Omega)$:
$$E(v) := \int_{\Omega} (W(\nabla v) + \psi(x, v)) \, dx$$
where $W$ is a convex energy density with two-sided $p$-growth, and $\psi$ is a convex lower-order term.

The Core Challenge:
While conforming finite element methods (FEM) for these problems are well-understood and provide optimal convergence rates for the error in energy (specifically $O(h^{2k})$ for polynomial degree $k$), non-conforming methods (such as Discontinuous Galerkin) have historically suffered from suboptimal convergence rates in the energy norm.

• Existing a priori error estimates for non-conforming schemes often yield rates like $O(h^{k/(p-1)})$, which are inferior to the conforming rates.
• This creates an imbalance in a posteriori error control based on the "two-energy principle" (comparing discrete and exact minimal energies), as the theoretical bounds do not match the observed performance or the conforming case.

2. Methodology

The authors propose a refined analysis using Local Discontinuous Galerkin (LDG) methods and Hybridizable methods, leveraging a novel discrete duality framework.

A. Discretization (LDG Scheme)

The method employs a discrete ansatz space $V_h$ of piecewise polynomials. A key component is the discrete gradient $\nabla_h v_h$, defined via a weak formulation involving lifting operators and jump terms across element faces. The discrete energy $E_h(v_h)$ includes a stabilization term $s_h(v_h)$ to control the jumps in the solution, ensuring coercivity and stability.

B. Discrete Duality Relations

The central innovation is the establishment of a strong duality relation between the discrete primal problem (minimizing $E_h$) and a semi-discrete dual problem (maximizing a dual energy $E^*_h$).

• Dual Space: The dual ansatz space $Y$ consists of pairs $(\tau_M, \tau_F)$ representing the flux and its normal traces.
• Duality Gap: The authors prove that for specific choices of stabilization parameters, the duality gap between the discrete primal minimum and the discrete dual maximum vanishes (or is negligible).
• Key Identity: They establish that $\max E^*_h(Y) = \min E_h(V_h)$ under mild regularity assumptions. This allows the energy error to be bounded by the difference between the interpolated exact solution and the discrete dual solution.

C. Error Analysis Strategy

1. A Priori Analysis: By utilizing a "conforming companion" operator $J_h$ (which maps non-conforming functions to conforming ones) and the duality relations, the authors derive error bounds.
   • They show that the energy error $|E(u) - \min E_h(V_h)|$ is bounded by terms involving the interpolation error of the exact solution and the stabilization terms.
   • Crucially, they prove that with appropriate choice of stabilization parameters (specifically the exponent $s$), the convergence rate for the energy error becomes quadratic ($O(h^{2k})$), matching the optimal rates of conforming FEM.
2. A Posteriori Control:
   • The paper constructs a conforming dual variable (using Raviart-Thomas finite elements) via explicit post-processing of the discrete dual solution.
   • This conforming dual variable provides a guaranteed upper bound on the energy error without solving local equilibrium problems, addressing the reliability-efficiency gap often found in residual-based estimators for degenerate problems.

3. Key Contributions

1. Optimal A Priori Rates for Non-Conforming Methods: The paper closes the theoretical gap between conforming and non-conforming discretizations for convex minimization. It proves that LDG methods achieve the same optimal energy convergence rates ($O(h^{2k})$) as conforming FEM for $p \ge 2$ (and improved rates for $1 < p < 2$).
2. Novel Duality Framework: It introduces a discrete duality relation that exploits the structure of the dual problem to bypass the suboptimal estimates typically associated with non-conforming gradients.
3. Guaranteed A Posteriori Error Bounds: The authors provide a constructive method to generate a conforming dual flux (Raviart-Thomas) that yields a strict upper bound on the energy error, enabling robust adaptive mesh refinement.
4. Extension to Hybridizable Methods: The analysis is extended to Hybridizable Discontinuous Galerkin (HDG) and Weak Galerkin (WG) methods, showing that the duality principles hold for these classes as well (with specific stabilization requirements).

4. Results

• Theoretical Convergence:
  • For $p \ge 2$ and smooth solutions, the energy error converges as $O(h^{2k})$.
  • For the stress variable (dual variable), the error converges as $O(h^{k})$ (or $O(h^{2k/p})$ in specific norms), improving upon previous suboptimal bounds like $O(h^{k/(p-1)})$.
  • The results hold for degenerate convex problems where the Hessian of $W$ may vanish.
• Numerical Benchmarks:
  • The authors tested the method on three problems: an Optimal Design problem, a 4-Laplace problem, and Bingham flow through a pipe.
  • Uniform Meshes: Showed suboptimal convergence rates (e.g., $O(h^{2/3})$) due to singularities in the solutions.
  • Adaptive Mesh Refinement (AMR): Using the proposed a posteriori error estimator, the adaptive algorithm successfully resolved singularities (e.g., at re-entrant corners or material interfaces).
  • Outcome: The adaptive strategy recovered optimal convergence rates (e.g., $O(\text{ndof}^{-k})$) for all polynomial degrees $k$, significantly outperforming uniform refinement.

5. Significance

This paper is significant because it resolves a long-standing theoretical limitation in the application of non-conforming finite element methods to nonlinear convex minimization problems.

• Theoretical Impact: It demonstrates that non-conforming methods are not inherently inferior to conforming methods in terms of energy convergence, provided a refined duality analysis is used.
• Practical Impact: The development of a guaranteed, computable a posteriori error estimator allows for reliable adaptive mesh refinement. This is crucial for solving complex engineering problems (like optimal material design or non-Newtonian fluid flow) where solutions often exhibit singularities or sharp transitions that uniform meshes cannot resolve efficiently.
• Robustness: The method handles degenerate convexity and $p$-Laplace type non-linearities effectively, making it a robust tool for a wide class of variational problems.

In summary, Carstensen and Tran provide a rigorous mathematical foundation and practical algorithmic framework that elevates Local Discontinuous Galerkin methods to the status of optimal solvers for convex minimization, matching the performance of conforming methods while retaining the flexibility of non-conforming discretizations.