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Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation

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Pith's one-line read Nondivergence-form PDEs with Cordes coefficients are equivalent to a well-posed ultraweak variational problem, and DPG discretizations of it converge quasi-optimally whenever a bounded Fortin operator exists.

desk verdict First ultraweak DPG treatment of Cordes nondivergence-form problems; the continuous theory is sound, but the fully discrete DPG scheme is proven only for piecewise-polynomial A — the DPG-LS variant covers the general case. Worth refereeing, with conditions. read the letter →

arxiv 1908.08981 v1 pith:F7WMGMGA submitted 2019-08-23 math.NA cs.NA

Thomas Führer This is my paper · ORCID
classification math.NAcs.NA MSC 65N3065N12
keywords ultraweakformulationDPGmethodnondivergenceformCordesconditionFortinoperatoraposteriorierrorestimationadaptivemeshrefinementtracespaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that a linear second-order PDE in nondivergence form, $Lu = A:D^2u = f$ with zero Dirichlet boundary data, can be rewritten as an ultraweak variational problem when the coefficient matrix $A$ satisfies the Cordes condition. The Hessian $D^2u$ becomes an independent $L^2$-valued unknown $M$, and the continuity of $u$ is carried by a trace variable, so no $H^2$-conforming discrete space is needed. The ultraweak problem is well-posed and equivalent to the original strong problem, and the DPG discretizations inherit quasi-optimal convergence with symmetric positive definite linear systems and local adaptive indicators. Fully discrete quasi-optimality is proven under a Fortin-operator condition; the paper verifies that condition when $A$ is piecewise polynomial of degree $p$, leaving the general Cordes case — including the radial-annulus example of Section 6.3 — as an explicit gap in the theory.

What carries the argument

The mechanism is the broken ultraweak bilinear form $b(u,v) = (u,-\operatorname{divDiv}Q)_T + (M,Av+Q) + \langle\hat u,Q\rangle_S$ on $U\times V$, together with the DPG trial-to-test operator $\Theta$. Elementwise integration by parts moves the second derivatives from $u$ onto the test flux $Q$ and produces the trace pairing $\langle\hat u,Q\rangle_S$; Proposition 3 identifies the trace norm with a supremum over $Q$, and Lemma 11 characterizes the subspace with vanishing normal-normal trace as the annihilator of the trace space. These identities feed the breaking-spaces theorem, which yields well-posedness. For the practical schemes, the Fortin operator $\Pi_F$ satisfying (10) is the load-bearing device: it preserves the bilinear form on the trial space and is bounded, making the discrete trial-to-test operator $\Theta_h$ a faithful replacement of $\Theta$.

What would settle it

Take a Cordes-coefficient problem with $A$ not piecewise polynomial on any uniformly refined mesh (for instance the radial three-annulus example of Section 6.3) and compare the discrete error ratio $\|u-u_h\|_U / \min_{w_h\in U_h}\|u-w_h\|_U$ along the adaptive sequence; if this ratio grows without bound while the estimator $\eta$ remains reliable, then the missing Fortin operator is essential rather than a technicality. Conversely, proving the existence of a bounded Fortin operator for every Cordes $A$ would settle the paper's main unresolved limitation in the affirmative.

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Extended reading notes

Core claim

The central claim, stated as Theorem 4 and Proposition 5, is that the strong problem (1) is equivalent to the ultraweak problem (7) and that (7) has a unique solution $u^\star\in U$ with $\|u^\star\|_U\le C\|f\|$. Every strong solution $u\in H^2(\Omega)\cap H^1_0(\Omega)$ maps to the ultraweak solution $(u,D^2u,\operatorname{tr}^2_T u)$, and every ultraweak solution has a first component in $H^2(\Omega)\cap H^1_0(\Omega)$ whose Hessian is $M$. For the DPG method, the paper proves quasi-optimality of the semi-discrete problem and of the fully discrete problem when a bounded Fortin operator with the Fortin property exists; Theorem 23 constructs such an operator for the lowest-order trial space when $A$ is piecewise polynomial of degree $p$. The paper also proves a posteriori error equivalence, $\|u-u_h\|_U^2\simeq\eta_{\mathrm{DPG}}^2$, under the same Fortin assumption.

Load-bearing premise

The fully discrete quasi-optimality bounds and the a posteriori equivalences presuppose a Fortin operator satisfying (10) (or (17)); the paper constructs one only when the coefficient matrix $A$ is piecewise polynomial of degree $p$, so for general Cordes coefficients, including the radial-annulus example of Section 6.3, the convergence theorems as stated do not apply.

Editorial extensions

If this is right

  • The original nondivergence equation can be approximated with discontinuous piecewise-polynomial spaces for both $u$ and the Hessian $M$, leaving only the trace of the solution to be discretized by H2-type trace elements.
  • The resulting DPG systems are symmetric positive definite, and the estimator $\eta_{\mathrm{DPG}}$ (equivalently $\eta_{\mathrm{LS}}$) is equivalent to the $U$-norm error whenever the Fortin assumption holds, so adaptive mesh refinement by local indicators is justified in that regime.
  • For piecewise-polynomial coefficient matrices aligned with the mesh, the fully discrete DPG and DPG–least-squares methods coincide, so the implementations and their error behavior are interchangeable.
  • Where the Fortin assumption is not verified, the quasi-optimal bounds (12) and (19) are not in force; Section 6.3 demonstrates that the two fully discrete schemes can then produce different approximations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the open issue is precisely the existence of a Fortin operator for non-piecewise-polynomial Cordes coefficients; if one were constructed, the quasi-optimality and reliability theorems would extend to the radial-annulus example, whose observed adaptive rates in Section 6.3 already match the desired behavior.
  • Editorial inference: the trace-space template imported from the Kirchhoff–Love plate problem suggests the same ultraweak-plus-trace construction may apply to other linear operators whose highest-order derivative enters linearly, such as fourth-order problems or systems with comparable coefficient structure.
  • Editorial inference: a direct construction of a Fortin operator for the DPG–least-squares scheme, without assuming $A$ is piecewise polynomial, would also yield one for the DPG method, because Section 2.7 shows the two schemes are equivalent at the operator level; commuting projections on $A$-weighted spaces are a natural starting point.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops an ultraweak variational formulation for linear elliptic PDEs in nondivergence form with Cordes coefficients. The formulation introduces the Hessian M=D^2u and a broken trace variable, leading to a well-posed problem in U=L2(Ω)×L2_sym(Ω)×Uhat. The paper proves equivalence with the strong problem, proposes DPG and DPG-least-squares discretizations, proves quasi-optimality under a Fortin assumption, constructs Fortin operators for lowest-order spaces, gives a posteriori error estimators with reliability and efficiency, and reports numerical experiments with uniform and adaptive refinement.

Significance. If the central claims hold, the paper is a useful contribution: it brings the DPG framework to Cordes-coefficient nondivergence-form problems, avoids H2-conforming elements by using only traces of rHCT elements, produces symmetric positive definite systems, and supplies local error indicators. The analysis is largely self-contained and the proofs of well-posedness and quasi-optimality are written out; the Fortin operator for the DPG-least-squares variant works for arbitrary Cordes coefficients because the form c in (14) is independent of A. The main weakness is a mismatch between the general Cordes-coefficient scope advertised in the abstract and the fully discrete convergence theory for the DPG method of Section 2.5 as implemented: that theory requires A to be piecewise polynomial on the mesh. This is a scope issue rather than a fundamental obstruction, and it can be repaired either by narrowing the claims or by proving a Fortin operator for general A (e.g., using the space N_h=A(M_h) suggested in Section 2.7).

major comments (2)
  1. [Sections 2.5, 5.2, 6.3; Theorems 7 and 14] The fully discrete convergence result for the DPG method of Section 2.5, Theorem 7, and the a posteriori equivalence (22) of Theorem 14 are conditional on the Fortin assumption (10). The only construction for the implemented method is Theorem 23, which assumes A∈P^p(T) for the chosen polynomial test space V_h^p=P^p×Q_h. Section 6.3 uses coefficients discontinuous along circles that are not aligned with the mesh, so A∉P^p(T) on the active triangulations; the paper itself states that the two methods are then no longer equivalent and that Theorem 23 does not hold. Consequently, Theorems 7 and 14 do not, as written, apply to the DPG method for the general Cordes regime advertised in the abstract, and the Section 6.3 DPG results are numerical evidence rather than proven convergence. Please either restrict the abstract and introduction to the case covered by Theorem 23, or add a Fortin construction for general Cordes coefficients (for instance by taking the first test component to be N_h=A(M_h) as in Section 2.7 and verifying (10) with Π_F=(Π_N_h, Π_divDiv)); the latter is straightforward in principle but is not carried out in the paper.
  2. [Section 3.2, Lemma 11 and Proposition 13] Lemma 11 is load-bearing for the proof of Theorem 4 because Proposition 13 uses it to identify the space V_0={v∈V : bhat(uh,v)=0 for all uh∈Uhat}. The proof is omitted and attributed to the author's own [14, Proposition 3.8] and [15, Proposition 11]. Since the cited statement is for traces of H^2_0(Ω) functions and the present setting uses traces of X=H^2(Ω)∩H^1_0(Ω), the transfer is not literally automatic. Please include the short adaptation argument or state precisely why the proof in [14] covers the present case; this will remove a nontrivial dependency from the central well-posedness theorem.
minor comments (3)
  1. [Proof of Theorem 14, Section 4] In the proof of Theorem 14, the identity (f,v−Π_L2 v)=(f−A:Q_h,v−Π_L2 v) is asserted for any Q_h∈L2_sym(Ω). The Fortin property (10) supplies this only for Q_h belonging to the discrete space M_h (the second component of U_h), since it is verified for trial functions of the form (0,M_h,0). The subsequent choice Q_h=M_h is admissible, so the argument is repairable, but the sentence as written overstates the range of Q_h and should be corrected.
  2. [Section 6, numerical examples] The text uses the notation 'Vh0' for the discrete test space; for consistency with Section 5.2 this should be V_h^0, and the polynomial degree p in V_h^p should be stated explicitly where the test spaces are first used in the experiments.
  3. [Section 2.7] The remark that a basis for N_h={A:M_h} is hard to determine is important for practice, but it would be helpful to add a sentence explaining that the theoretical Fortin construction with N_h is nonetheless valid for any bounded A, so that the limitation is one of implementation rather than of the ultraweak formulation itself.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central well-posedness and quasi-optimality arguments use independent inf-sup, breaking-space, and Fortin-operator results, while the only self-cited ingredients are parameter-free trace lemmas that do not encode the target result.

full rationale

The derivation chain is not circular. Theorem 4 is obtained by checking the hypotheses of the external breaking-spaces theorem [3, Theorem 3.3]: Proposition 12 proves the two-field inf-sup condition by a self-contained adjoint-problem argument (Lemmas 2 and 10), and Proposition 13 identifies the kernel of the trace form. The trace ingredients used there (Proposition 3 and Lemma 11) are quoted without proof from the author's own prior work [14, 15]; they are load-bearing for the trace-space identification, but they are parameter-free statements about H(div Div) traces and H2 trace spaces, with no reference to the Cordes coefficient A, the PDE, or any fitted data. Under the independence rule, such cited results are real evidence rather than circularity: they do not include the target result. The fully discrete DPG quasi-optimality theorems (Theorems 7, 9, 14, 16) are explicitly conditional on the existence of Fortin operators (10)/(17), and the paper constructs these operators under stated hypotheses in Lemma 21 and Theorem 23. The paper itself flags the one notable coverage gap: in Section 6.3 'the discontinuities of the coefficients are not aligned with the mesh. Therefore, both methods are not equivalent (Theorem 23 does not hold).' This is an honest limitation of the proof for general Cordes coefficients in that example, not a definitional or fitted-input circularity. No parameter is fitted and then renamed a prediction; the a posteriori estimators bound residuals and are compared against independently computed errors. The overall derivation therefore does not reduce to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. Its analysis rests on the Cordes/domain assumptions, the author's prior trace theory (not re-proven here), and a Fortin-existence assumption verified only for piecewise polynomial coefficients.

assumptions (5)
  • domain assumption The coefficient matrix A satisfies the Cordes condition (2c) with parameter 0 < ε ≤ 1 and uniform ellipticity bounds (2b).
    This is the problem class; the inf-sup analysis in Lemma 2 and Lemma 10 depends on the Cordes inequality.
  • domain assumption The domain Ω is bounded, convex, polytopal, and Δ maps X = H² ∩ H¹_0 onto L²(Ω).
    Used to ensure strong solutions and the norm equivalence ||D²v|| ≃ ||Δv|| (Section 2.2, Proposition 1).
  • domain assumption Trace space results from the author's prior works: Proposition 3 (dual norm characterization) and Lemma 11 (trace kernel H_0(divDiv)) are stated without proof, citing [14, Prop 3.8/3.9] and [15].
    These are load-bearing for Theorem 4 via the breaking-spaces argument (Section 3.3); if false, the well-posedness proof collapses. The paper does not reproduce the proofs.
  • standard math The 'breaking spaces' theorem [3, Theorem 3.3] is used without proof to conclude Theorem 4.
    A published abstract framework for DPG well-posedness; the paper checks its hypotheses (Prop 12, Prop 13).
  • ad hoc to paper Existence of a Fortin operator for general A: assumptions (10)/(17) are imposed; only verified when A ∈ P^p(T) (Theorem 23).
    The fully discrete convergence theorems are conditional on this assumption; for non-polynomial or mesh-unfitted coefficients, the assumption is unproven.

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Pith. "Pith review of Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation." pith.science (2026). https://pith.science/paper/F7WMGMGA

@misc{pith2026190808981,
  author       = {Pith},
  title        = {Pith review of: Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7WMGMGA}},
  note         = {Machine review of arXiv:1908.08981}
}
read the original abstract

We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov--Galerkin (DPG) methods. We investigate Fortin operators for the fully discrete schemes and provide a posteriori estimators for the methods under consideration. Numerical experiments are presented in the case of uniform and adaptive mesh-refinement.

Figures

Figures reproduced from arXiv: 1908.08981 by the authors.

Figure 1
Figure 1. Initial triangulation of Ω = (−1, 1)2 with 16 elements. This example gives numerical evidence that higher convergence rates are possible for one solution component by increasing the polynomial degree. 6.1. Example with regular solution. We consider the problem from [22, Section 6.1], where A = A(x, y) =  2 sign(xy) sign(xy) 2  and f is chosen such that u(x, y) =  xe1−|x| − x  ye1−|y| − y  is the exact solution… view at source ↗
Figure 2
Figure 2. Errors of field variables and estimators for the example from Section 6.1. 102 103 104 10−1 100 1 3 1 2 number of degrees of freedom ku − uhk kM − Mhk ηDPG 102 103 104 10−1 100 1 2 1 2 number of degrees of freedom ku − uhk kM − Mhk ηLS [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. L 2 field errors and error estimator for the example from Section 6.2 The left plot shows the results on a sequence of uniform refinements and the right plot shows the results in the case of adaptive refinements. the estimator for the method from Section 2.5. Using an adaptive strategy the optimal rates are recovered, see again [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Meshes generated by adaptive algorithm for the example from Section 6.2. where g(x, y) =    1 if 0 ≤ p x 2 + y 2 < 1/3, −1 if 1/3 < p x 2 + y 2 < 2/3, 0 else . In this case an exact solution is not known. As can be observed from [PITH_FULL_IMAGE:figures/full_fig_…
Figure 5
Figure 5. Figure 5: Energy error for the example from Section 6.3. The idea of using augmented trial spaces in DPG methods based on ultraweak formulations stems from the author’s recent works [10, 11]. There it was shown, under some standard regularity assumptions, that the use of augment…
Figure 6
Figure 6. Figure 6: Sequence of adaptively generated meshes and corresponding solution component uh for the example from Section 6.3. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Approximation of the Hessian M = D2u, i.e., Mh,11 (left), Mh,12 (mid￾dle), Mh,22 (right) on a sequence of adaptively generated meshes for the problem from Section 6.3. [15] T. Führer, N. Heuer, and F.-J. Sayas. An ultraweak formulation of the reissner–mindlin plate ben…
Figure 8
Figure 8. Figure 8: Errors of field variables and estimators for the example from Section 6.4. [23] I. Smears and E. Süli. Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equa￾tions with Cordes coefficients. SIAM J. Numer. Anal., 52(2):993–1016, 2014. [24] I…

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