DPG-Bench: DPG Method Benchmark Suite

• DPG-Bench is a framework that benchmarks and analyzes the discontinuous Petrov–Galerkin method for solving diverse PDEs.
• It integrates rigorous error analysis, optimal test space construction, and stable discretizations to ensure reliable numerical approximations.
• DPG-Bench supports benchmark problems like Laplace, Helmholtz, and Maxwell equations, with scalable preconditioning and adaptive refinement strategies.

DPG-Bench is a collective term referring to the theoretical, algorithmic, and practical apparatus for benchmarking, analyzing, and implementing the Discontinuous Petrov-Galerkin (DPG) method and its variants across a diversity of partial differential equations (PDEs), numerical formulations, and computational regimes. Developed through a series of foundational and recent works by J. Gopalakrishnan, L. Demkowicz, and collaborators, DPG-Bench encapsulates the rigorous error analysis, robust implementation strategies, and reference problem suites essential for both research and applied evaluation of DPG methods.

1. Foundational Principles of the DPG Method

The DPG method is a finite element framework in which stability and reliability are achieved by computing optimal test spaces for a given trial space. At its core, DPG minimizes the residual of the governing PDE in a user-chosen norm—typically by means of a trial-to-test operator $T$:

$(T w, v)_V = b(w, v), \quad \forall v \in V$

where $b(\cdot, \cdot)$ is the bilinear form corresponding to the weak (variational) formulation, and $V$ is the test space. A key outcome of this construction is that DPG discretizations automatically satisfy a discrete inf-sup (stability) condition, guaranteeing quasi-optimality:

$$\|u - u_h\| \leq C \inf_{w_h \in U_h} \| u - w_h \|$$

for the exact solution $u$, DPG approximation $u_h$, and trial space $U_h$ (1107.4293).

Practical DPG replaces infinite-dimensional test functions with enriched finite-dimensional polynomial spaces, selecting test degree $r$ at least $p + N$ (where $p$ is the trial polynomial degree and $N$ the spatial dimension), to retain optimal rates.

2. Critical Components and Error Analysis

DPG-Bench emphasizes rigorous error analysis and practical implementation guidance drawn from several model problems:

• Ultraweak Formulations: All derivatives are transferred onto the test functions, supporting discontinuous (broken) test spaces and local test function computation. This formulation underlies DPG's element-wise stability and efficiency in assembling the system matrix (1306.0557, 1507.05428).
• Fortin Operators: Existence and construction of Fortin operators is pivotal for ensuring reliability when test spaces are approximated (1107.4293). This enables the passage from "ideal" to "practical" DPG without loss of stability or error control.
• A Priori Estimates: Using enriched test spaces and the ultraweak setting, DPG methods achieve, for smooth solutions,

$$\| \sigma - \sigma_h \|_{L^2}^2 + \| u - u_h \|_{L^2}^2 + \|\dots\| \leq C h^{2s}$$

for suitable $s \leq p+1$. Failure to enrich sufficiently (i.e., $r < p + N$) can induce suboptimality.

3. Benchmarks, Model Problems, and Methodological Variants

DPG-Bench consists of a growing suite of models for benchmarking theory and software, including but not limited to:

• Laplace and Linear Elasticity: Classic benchmarks, confirming optimal convergence and locking-freeness in the presence of homogeneous isotropic materials (1107.4293).
• Helmholtz Equation: Dispersion- and dissipation-focused benchmarks, utilizing scaled test norms of the form

$\|v\|_V^2 = \|A_h v\|^2 + \varepsilon^2 \|v\|^2$

to reduce artificial dissipation and improve accuracy for wave problems. The parameter $\varepsilon$ tunes the balance, with vanishing $\varepsilon$ yielding near-best approximation, but requiring careful discrete test function enrichment (1304.7497).

• Maxwell Equations: Stability inherited automatically across strong, ultraweak, and mixed formulations, with proven quasi-optimality and error estimator equivalence (1507.05428).
• Eigenvalue Problems: Primal and ultraweak DPG discretizations for the Laplace eigenproblem, with optimal convergence and a priori as well as a posteriori error estimators suitable for adaptive schemes (2012.06623).
• Quad-Curl, Plate Bending, and Stokes: Robust handling of higher-order PDEs, supporting minimal regularity and pressure-robust formulations via curl elimination (2301.10588).
• Spacetime Problems: Fully spacetime-coupled DPG discretizations enabling unstructured adaptivity in time and space (1709.08268).

4. Implementation, Preconditioning, and Scalability

DPG-Bench encompasses practical frameworks for assembling, preconditioning, and solving the DPG linear systems:

• System Conditioning: DPG stiffness matrices are symmetric positive definite, with condition numbers scaling as $\mathcal{O}(h^{-2})$, unaffected by polynomial degree for shape-regular, quasi-uniform meshes (1107.4293).
• Multigrid and AMG Preconditioners: Both geometric and algebraic multigrid preconditioners have been developed specifically for DPG matrices, including Schur complement approaches on interface degrees of freedom and additive Schwarz smoothers. Trace-based multigrid and static condensation are key for handling large-scale, adaptive and high-order computations efficiently (1608.02567, 1612.00838, 2010.06793).
• Adaptivity: Intrinsic, localizable a posteriori error estimation is built into the DPG framework via the error representation function, supporting robust $h$- and $p$-adaptive refinement (1306.0557, 2012.06623).

5. Robustness, Limitations, and Methodological Advances

Recent benchmark analyses have established critical limitations and corresponding remedies:

• Domain and Poincaré Locking: On large domains or with boundary conditions that weaken the underlying Poincaré inequality, standard DPG test norms lead to instability and convergence "locking" (i.e., poor or vanishing error reduction). This effect is more pronounced for higher-order PDEs (2011.12098). Uniform robustness is restored by scaling test norms to account for domain size:

$$\|v\|_{1, d}^2 = d^{-2} \|v\|_{L_2}^2 + \|\nabla v\|_{L_2}^2$$

where $d$ reflects the domain-dependent Poincaré constant.

• Comparison to Alternative Methods: DPG with small test norm scaling parameter ($\varepsilon$) outperforms least squares in dispersion and dissipation metrics for Helmholtz-like problems, though all methods are subject to pollution error (1304.7497).
• Couplings and Extensions: DPG is flexibly coupled to standard finite element schemes (FEM) for heterogeneous domain decompositions, allowing robust discretization selectively where needed (1704.07471).

6. Practical Benchmarking Considerations

DPG-Bench provides a reference for implementing, testing, and comparing DPG discretizations:

• Test Norm Selection: Benchmarks must incorporate standard and scaled test norms, particularly in situations with large or anisotropic domains, or where the essential boundary for the Poincaré inequality is minimal, as failure to scale can mask inherent DPG robustness (2011.12098).
• Parameter Variation: Sensitivity to polynomial enrichment ($r$), scaling parameters ($\varepsilon$), and mesh properties should be systematically tested, including non-uniform and adaptive meshes, and a range of PDE models (elliptic, parabolic, hyperbolic, high-frequency wave).
• Preconditioning and Solver Performance: Benchmarks should report iteration counts, condition numbers, and parallel scalability, including the use of trace-space multigrid and algebraic multigrid preconditioners to enable large-scale adaptive computations (1608.02567, 1612.00838, 2010.06793).
• Adaptive Strategies: DPG-Bench frameworks routinely use built-in a posteriori error indicators to guide mesh adaptation and verify that adaptivity recovers optimal convergence in singular or complex domains (2012.06623, 1709.08268).

7. Summary Table: Key Aspects and Implications

Aspect	DPG-Bench Guidance	Practical Implication
Robustness	Requires scaled test norms on large domains	Essential for domain-independence
Preconditioning	Custom multigrid and trace-space AMG	Critical for large-scale computation
Error indicators	Intrinsic, localizable, reliable	Enables automated adaptive meshing
Eigenvalue/Sing. problems	Adaptive methods validated	Optimal rates in challenging regimes
Coupling/Hybrid	Interfaced DPG-FEM discretizations	Flexible for multiphysics applications

8. Conclusion

DPG-Bench comprises the theoretical, methodological, and computational blueprints necessary for robust and reproducible evaluation of DPG schemes across model PDEs, domains, and computational architectures. Carefully chosen test norms, sufficient test space enrichment, and attention to domain-induced stability challenges are paramount. Integrated adaptive refinement and preconditioning strategies are essential for scalable, reliable DPG performance on contemporary benchmarks in science and engineering.