---
title: Ultraweak Formulations
url: https://www.emergentmind.com/topics/ultraweak-formulations
type: topic
---

# Ultraweak Formulations

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 [2309.00726], [1706.06754], [1908.08981], [1805.07835].

## 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 [1706.06754], [2309.00726], [1805.07835].

## 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 [2309.00726], [1706.06754], [1612.03124], [1805.07835].

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 [2309.00726], [1805.07835].

## 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) [2309.00726], [1706.06754], [1612.03124], [1908.08981]. 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 [2309.00726].

| 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 [2309.00726], [1706.06754], [1908.08981], [2310.19674], [1805.07835].

## 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 [1706.06754].
- Linear PDEs in nondivergence form under Cordes condition [1908.08981].
- Least-squares ultraweak discretizations for unique continuation and Cauchy problems, measuring data and consistency in weaker norms and supporting nonconforming test spaces [2407.04571].
- Time-dependent Schrödinger and DAE problems, admitting model reduction by optimal-in-time reduced basis methods [2202.12834], [1610.04678], [2310.19674].
- Plate bending problems for Reissner–Mindlin and Kirchhoff–Love models, resolving locking issues [1906.04869], [1805.07835].
- Viscoelastic fluid models, ensuring inherent stability without additional stabilization mechanisms [1612.03124].
- Hypersingular integral equations with direct control over singular and discontinuous features [1309.1697].

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.

## References

- "An Anisotropic $hp$-Adaptation Framework for Ultraweak Discontinuous Petrov-Galerkin Formulations" [2309.00726]
- "High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations" [1706.06754]
- "Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation" [1908.08981]
- "Ultra-weak least squares discretizations for unique continuation and Cauchy problems" [2407.04571]
- "A spacetime DPG method for the Schrodinger equation" [1610.04678]
- "An ultraweak variational method for parameterized linear differential-algebraic equations" [2202.12834]
- "An ultraweak DPG method for viscoelastic fluids" [1612.03124]
- "An ultraweak formulation of the Reissner-Mindlin plate bending model and DPG approximation" [1906.04869]
- "An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation" [1805.07835]
- "Ultra-weak formulation of a hypersingular integral equation on polygons and DPG method with optimal test functions" [1309.1697]
- "Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems" [2310.19674]
- "Trace operators of the bi-Laplacian and applications" [1904.07761]

Source: https://www.emergentmind.com/topics/ultraweak-formulations