Ultraweak Formulations

• Ultraweak formulations are advanced variational methods that shift all differential operators from the trial space to the test space via repeated integration by parts.
• They employ discontinuous trial functions with weak coupling enforced by skeleton unknowns on mesh interfaces, ensuring stability and proper boundary enforcement.
• These formulations underpin optimally stable DPG methods that include built-in a posteriori error estimators and support highly adaptive hp-refinement strategies.

Ultraweak formulations are advanced variational constructs wherein all differential operators are transferred from the trial space onto the test space via repeated local integration by parts. This results in trial spaces composed entirely (or almost entirely) of $L^2$-type functions, with weak interelement coupling enforced by skeleton (trace) unknowns defined on mesh interfaces. The ultraweak framework enables the construction of robust and optimally stable Petrov-Galerkin and least-squares discretizations, featuring built-in a posteriori error estimators and facilitating highly adaptive $hp$-refinement strategies. Ultraweak formulations generalize standard and mixed weak forms to wider classes of PDEs, including elliptic, parabolic, hyperbolic, and nondivergence form operators, and are foundational in the development of discontinuous Petrov-Galerkin (DPG) methods.

1. Mathematical Construction and Fundamental Principles

In an ultraweak variational formulation, derivatives in the model equations are shifted entirely onto the test functions. For the Poisson equation in first-order form on $\Omega\subset \mathbb{R}^d$,

$$\sigma - \nabla u = 0,\qquad -\nabla\cdot\sigma = f,$$

the ultraweak DPG formulation utilizes broken trial and test spaces over a mesh partition $\Omega_h$. The typical settings are:

• Trial space: $U = L^2(\Omega)\times[L^2(\Omega)]^d$.
• Skeleton (interface/trace) unknowns: $\hat{U}$, e.g., $H^{1/2}(\Gamma_h)\times H^{-1/2}(\Gamma_h)$.
• Test space: $V=H^1(\Omega_h)\times H(\mathrm{div},\Omega_h)$.

Such construction shifts all regularity requirements onto the test space, thus allowing discontinuous, nonconforming, and piecewise polynomial trial functions with weak enforcement of interelement compatibility via skeleton traces (Chakraborty et al., 2023, Astaneh et al., 2017, Führer, 2019, Führer et al., 2018).

2. Skeleton Variables and Weak Coupling

A distinguishing feature of the ultraweak approach is the enforcement of interelement coupling and boundary conditions using skeleton variables. These quantities, such as $\hat{u}\in H^{1/2}(\Gamma_h)$ and $\hat{\sigma}_n\in H^{-1/2}(\Gamma_h)$, represent traces (jumps) of the primal and flux variables on the mesh skeleton. Their role is to facilitate weak compatibility, guarantee well-posedness, and encode Dirichlet or Neumann boundary constraints. All interelement continuity is mediated by duality pairings in the bilinear form, not by global continuity constraints in the trial space (Astaneh et al., 2017, Chakraborty et al., 2023, Führer et al., 2018).

3. Optimal Test Spaces and Discontinuous Petrov-Galerkin

The effectiveness of ultraweak formulations is maximized in the DPG framework, which constructs optimal test functions for each trial basis function. For every trial $\psi_j$, one solves locally for $t_j\in V$ such that

$(t_j, w)_V = b(\psi_j, w),\quad\forall w\in V,$

which attains the supremum over the test space—these “supremizers” ensure inf-sup stability and minimize the residual in the energy norm. The construction is localized due to the broken test spaces, and the Gram matrix $G_K$ for each element is invertible, resulting in robust elementwise solves (Chakraborty et al., 2023, Astaneh et al., 2017, Keith et al., 2016, Führer et al., 2018).

The key mechanism is that the energy norm induced by the DPG method, based on the adjoint graph norm of the test space, is equivalent to an $L^2$-norm of the trial variables for many model problems. This equivalence guarantees orthogonal projection (best approximation) and reliable error estimation (Chakraborty et al., 2023, Führer et al., 2018).

4. Built-in A Posteriori Error Estimation

Ultraweak DPG schemes derive a natural, residual-based a posteriori error estimator. Via the Riesz map $R_V: V \to V^\prime$,

$$\|\text{true error}\|_{E, K} = \|R_V^{-1}(\ell_K - b_K(u_h, \cdot) - \hat{b}_K(\hat{u}_h, \cdot))\|_{V(K)},$$

where $\ell_K$, $b_K$, and $\hat{b}_K$ are local load, stiffness, and trace stiffness matrices. Summing these over elements yields a global error norm that is both reliable (upper bound of the true error) and efficient (comparable to the true error up to data oscillation) (Chakraborty et al., 2023, Astaneh et al., 2017, Keith et al., 2016, Führer, 2019). This estimator supports fully automatic mesh refinement and drives adaptive strategies.

5. Adaptive $hp$-Refinement Algorithms

Ultraweak formulations, combined with DPG error estimation, facilitate advanced anisotropic $hp$-adaptation. The framework involves:

• Computing local residual indicators $\{\eta_K\}$.
• Marking elements via Dörfler strategy (e.g., set $M$ with $\sum_{K\in M}\eta_K^2 \geq \theta\sum_K \eta_K^2$, $\theta$ a user parameter).
• Generating a fine reference mesh by isotropic $hp$-refinement.
• For each marked coarse element, staging a local competition among anisotropic $h$-, $p$-, and $hp$-refinements based on guaranteed error reduction rates: $$e_\mathrm{hp}(S_{\mathrm{new}}) = \frac{\|\text{ref} - P_\mathrm{old}\text{ref}\|^2_{L^2(K)} - \|\text{ref} - P_\mathrm{new}\text{ref}\|^2_{L^2(K)}}{\mathrm{ndof}_\mathrm{new} - \mathrm{ndof}_\mathrm{old}},$$ where the optimal refinement is chosen to maximize $e_\mathrm{hp,K}^*$.

This methodology leads to exponential convergence rates in total degrees of freedom for analytic solutions and greatly outperforms standard isotropic refinement schemes, particularly in the presence of boundary and internal layers (Chakraborty et al., 2023).

Step	Description
Mark	Use $\{\eta_K\}$ and Dörfler threshold to select mesh elements
Refine	Isotropically $hp$-refine marked elements for reference solve
Compete	For each coarse marked element, test 8 anisotropic $h$-refinement, 1 $p$-refinement via $e_\mathrm{hp}$
Select	Apply refinement with guaranteed error reduction rate

6. Well-Posedness, Norm Equivalence, and Convergence Theory

The ultraweak DPG framework satisfies robust well-posedness (continuity and inf-sup stability) for a wide class of PDEs, including elliptic, nondivergence, higher-order, and time-dependent problems. The essential properties are:

• The DPG energy norm is equivalent to the relevant $L^2$-norm(s) of the primal variables.
• The built-in residual estimator is both reliable and efficient.
• For quasi-uniform meshes and fixed $p$, convergence rates of $O(h^p)$ are attained.
• $hp$-adaptive algorithms yield exponential convergence in the number of degrees of freedom for analytic solutions.
• Extends to geometric settings (polygonal elements, non-convex domains) and equations with parameter dependence (Chakraborty et al., 2023, Astaneh et al., 2017, Führer, 2019, Engwer et al., 2023, Führer et al., 2018).

7. Applications and Generalizations

Ultraweak formulations have been successfully applied to:

• Polygonal high-order finite elements and PolyDPG methods, where trial fields are discontinuous and skeleton unknowns encode all compatibility (Astaneh et al., 2017).
• Linear PDEs in nondivergence form under Cordes condition (Führer, 2019).
• Least-squares ultraweak discretizations for unique continuation and Cauchy problems, measuring data and consistency in weaker norms and supporting nonconforming test spaces (Monsuur et al., 5 Jul 2024).
• Time-dependent Schrödinger and DAE problems, admitting model reduction by optimal-in-time reduced basis methods (Beurer et al., 2022, Demkowicz et al., 2016, Engwer et al., 2023).
• Plate bending problems for Reissner–Mindlin and Kirchhoff–Love models, resolving locking issues (Führer et al., 2019, Führer et al., 2018).
• Viscoelastic fluid models, ensuring inherent stability without additional stabilization mechanisms (Keith et al., 2016).
• Hypersingular integral equations with direct control over singular and discontinuous features (Heuer et al., 2013).

A characteristic property in these settings is that ultraweak formulations permit discontinuous, non-smooth, even piecewise or localized trial approximations, yet the inf-sup stability and a posteriori error estimation are retained via precisely constructed broken test spaces and skeleton unknowns.

• "An Anisotropic $hp$-Adaptation Framework for Ultraweak Discontinuous Petrov-Galerkin Formulations" (Chakraborty et al., 2023)
• "High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations" (Astaneh et al., 2017)
• "Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation" (Führer, 2019)
• "Ultra-weak least squares discretizations for unique continuation and Cauchy problems" (Monsuur et al., 5 Jul 2024)
• "A spacetime DPG method for the Schrodinger equation" (Demkowicz et al., 2016)
• "An ultraweak variational method for parameterized linear differential-algebraic equations" (Beurer et al., 2022)
• "An ultraweak DPG method for viscoelastic fluids" (Keith et al., 2016)
• "An ultraweak formulation of the Reissner-Mindlin plate bending model and DPG approximation" (Führer et al., 2019)
• "An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation" (Führer et al., 2018)
• "Ultra-weak formulation of a hypersingular integral equation on polygons and DPG method with optimal test functions" (Heuer et al., 2013)
• "Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems" (Engwer et al., 2023)
• "Trace operators of the bi-Laplacian and applications" (Führer et al., 2019)