m-DPG: Modified DPG Finite Element Method

• m-DPG is a modified Discontinuous Petrov–Galerkin method that uses a scaled test norm with a tunable parameter to improve stability and convergence for wave-dominated problems.
• The method explicitly links the scaling parameter to the control of dispersive and dissipative errors, resulting in quasi-optimal error estimates and enhanced accuracy in phase and amplitude.
• It also supports mixed formulations and adaptive error control, making it effective for Helmholtz equations, transmission problems, and singularly perturbed applications.

The term m-DPG most commonly refers to the “modified DPG” method, a family of discontinuous Petrov–Galerkin (DPG) finite element discretizations in which the test space norm is altered—often via the inclusion of a tunable scaling parameter—to enhance stability, quasi-optimal convergence, and accuracy, particularly for wave-dominated and singularly perturbed problems. While sometimes “m-DPG” denotes “mixed DPG” (where the method is recast in a mixed or saddle-point formulation), the principal usage aligns with the “modified” norm approach, as in the analysis of dispersive and dissipative errors for Helmholtz-type equations (Gopalakrishnan et al., 2013).

1. Definition and Foundational Principles

The m-DPG method arises from a modification of the standard DPG approach, in which the test space is endowed with a scaled graph norm:

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

where $A_h$ is a (local) differential operator associated with the PDE (e.g., the Helmholtz operator), and $\varepsilon > 0$ is a user-defined scaling parameter (Gopalakrishnan et al., 2013). The usual DPG method corresponds to $\varepsilon=1$. When $\varepsilon \ll 1$, the method increasingly weights the derivative term, which can regularize and improve certain properties of the discretization.

The construction is embedded in a wider DPG framework (Gopalakrishnan, 2013), which selects test functions optimally (via a “trial-to-test” operator) to ensure discrete stability reflected in the preservation of the inf-sup constant from the continuous problem (Babuška–Brezzi stability). The m-DPG approach introduces further flexibility by modifying the test norm, with direct impact on error constants and the numerical spectrum.

2. Impact of the Scaling Parameter and Error Properties

One of the central results of the m-DPG methodology is the explicit linkage between the scaling parameter $\varepsilon$ and the dispersive/dissipative character of numerical solutions for wave problems (Gopalakrishnan et al., 2013):

• Dispersive errors arise when the real part of the computed discrete wavenumber $k_h$ deviates from the exact wavenumber $k$; dissipative errors are associated with a nonzero imaginary part of $k_h$, introducing artificial damping into propagating waves. Both are collectively referred to as “pollution” errors.
• Quantitative measures used in the analysis are:

$$\rho = \max_\theta \left| \operatorname{Re} k_h(\theta) - k \right|, \qquad \eta = \max_\theta \left| \operatorname{Im} k_h(\theta) \right|,$$

computed as functions of $\varepsilon$ and other mesh/test parameters.

• As $\varepsilon \to 0$, the quasioptimality constant for error estimates improves (i.e., approaches unity), leading to better error bounds for fluxes and traces. Specifically, the theoretical error estimate is:

$$\|u - u_h\|_U \leq (1 + c) \inf_{w \in U_h} \|u - w\|_U,$$

where $c$ is explicitly decreasing in $\varepsilon$ [(Gopalakrishnan et al., 2013), Theorem 4.1].

However, the benefit of small $\varepsilon$ can be compromised if the “local” test problem is solved inexactly (e.g., using an enrichment degree $r$ in the polynomial basis for test functions), especially when $r$ is even. Some apparent “discrete effects” (numerical artifacts) may emerge for very small $\varepsilon$ unless $r$ is chosen appropriately (often odd $r$ yields more favorable behavior).

3. Dispersion Analysis and Stencil Structure

The m-DPG method’s error and accuracy are deeply influenced by the interplay between mesh structure, stencil type, and test space enrichment (Gopalakrishnan et al., 2013):

• In a uniform mesh, static condensation of interior (elemental) degrees of freedom leaves a global system with distinct types of “interface” unknowns: vertices (scalar field), horizontal edges (vertical flux), and vertical edges (horizontal flux).
• The resulting stencil equations—formed by applying a Fourier plane wave ansatz—are summarized as $F(k_h)a = 0$, with the wavenumber $k_h$ numerically identified as the root of $\det F(k_h) = 0$ for each propagation angle.
• This analysis evidences that both dispersive and dissipative errors reduce as $\varepsilon$ decreases, provided the approximation of local test functions is sufficiently accurate.

An important conclusion is that standard DPG formulations (or classical $L^2$ least-squares methods) may exhibit significant amplitude damping. The m-DPG approach, with optimized $\varepsilon$, produces numerical plane waves much closer in phase and amplitude to analytical solutions.

4. Comparison with Alternative Methods

m-DPG offers distinctive advantages and trade-offs compared to standard discretizations:

• $L^2$ least-squares: The standard $L^2$ least-squares scheme exhibits much heavier dissipation, as demonstrated visually and quantitatively for plane wave problems (Gopalakrishnan et al., 2013).
• Higher-order finite elements: In some cases, very high-order finite elements (e.g., biquadratics) may outperform m-DPG in dispersion, but m-DPG retains advantages in Hermitian positive-definiteness, built-in stabilization, and error control—especially on coarse meshes or for flux/traces.
• Conventional DPG ($\varepsilon=1$): Both $L^2$ least-squares and the unmodified DPG approach ($\varepsilon=1$) fare worse in amplitude preservation than the tuned m-DPG method.

These comparisons indicate that m-DPG is particularly effective in regimes where mesh refinement is limited, the frequency is high, or stabilization against damping is required.

5. Numerical Behavior and Practical Guidelines

Numerical experiments reveal that:

• Smaller $\varepsilon$ leads to smaller dispersive and dissipative errors, up to a threshold determined by the accuracy of test space discretization (Gopalakrishnan et al., 2013).
• For $\varepsilon$ in the range $10^{-6}$, computed plane waves may become nearly indistinguishable from analytical solutions, even at moderate mesh resolution and high wave numbers.
• The convergence rate in wavenumber error can improve from quadratic ($O(h^2)$) to cubic ($O(h^3)$) as $\varepsilon$ is reduced, but the improvement is sensitive to enrichment degree $r$ of the test space.

For practical deployment:

• Select $\varepsilon$ as small as numerically feasible, constrained by the conditioning and discretization of local test problems.
• Prefer odd test space enrichment degrees $r$, as these generally yield more monotonic error improvement with decreasing $\varepsilon$.

6. Mixed and Residual-Minimization Formulations

The “mixed” DPG (m-DPG) perspective (Gopalakrishnan, 2013) highlights a saddle-point formulation in which an error representation function is introduced, yielding a system:

$$\begin{align*} (\epsilon, y)_Y + b(x_h, y) &= \ell(y) \quad \forall y \in Y, \ b(z_h, \epsilon) &= 0 \quad \forall z_h \in X_h, \end{align*}$$

where $\epsilon$ measures the residual in a dual norm. The method is equivalent to residual minimization and allows intrinsic, local a posteriori error indicators.

This formulation, often called m-DPG in the literature, offers a unifying framework for understanding both the original and modified DPG family, including robust adaptive mesh refinement strategies based on the norm of $\epsilon$.

7. Extensions and Applications

m-DPG variants and their analytical insights have been extended to a range of problems:

• Helmholtz and wave equations: primary domain of dispersive/dissipative analysis for m-DPG (Gopalakrishnan et al., 2013).
• Transmission problems: coupling of ultraweak interior DPG discretizations with boundary integral equations, ensuring optimal stability and convergence (Heuer et al., 2014).
• Singularly perturbed and reaction-dominated problems: where modifications to test norms (and construction of parameter-robust Fortin operators) substantially enhance robustness and stability (Führer et al., 2023).
• Locking-free solid mechanics: ultraweak and DPG/m-DPG formulations for Timoshenko and Euler–Bernoulli beams, with uniform convergence as thickness vanishes (Führer et al., 2020).

A plausible implication is that the m-DPG strategy—viewed as test norm modification to optimize error constants and adapt to problem structure—can be imported into more general adaptive, minimal, or multigrid-accelerated frameworks.

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In summary, m-DPG denotes a class of DPG methods distinguished by a scaled test norm, yielding significant practical advantages in stability, accuracy, and adaptivity for wave and singularly perturbed problems. Key features include explicit error estimates linked to the scaling parameter, systematic control over dispersive/dissipative errors, and natural compatibility with adaptive algorithms and mixed formulations. The effectiveness of m-DPG relies on judicious choice of the scaling parameter and careful design of enriched test spaces for numerical implementation.