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Breaking spaces and forms for the DPG method and applications including Maxwell equations
C. Carstensen, L. Demkowicz, J. Gopalakrishnan
TL;DR
DPG methods use broken test spaces to localize computations, but their analysis requires a precise link between broken and unbroken Sobolev spaces. This paper characterizes the interface spaces and transfers stability to broken formulations, yielding Maxwell error analysis and stability equivalences across formulations, while leaving explicit polynomial-degree and wavenumber dependence unresolved.
Problem
DPG analysis needs a precise characterization of interface spaces linking broken Sobolev spaces to their unbroken counterparts and a way to transfer stability between formulations.
Method
The paper characterizes interface spaces by duality, develops Fortin operators, and derives broken-form stability from corresponding unbroken-form stability.
Results
The framework provides complete DPG error analysis for time-harmonic Maxwell equations and shows that stability of one formulation implies stability of five others.
Takeaways & Limitations
Broken DPG schemes inherit stability from the exact equations and remain stable on arbitrarily coarse meshes, supporting robust adaptive meshing strategies.
Takeaways & Limitations
The analysis does not provide convergence results explicit in polynomial degree p or track dependence on the wavenumber ω.
Abstract
from arXiv · showhide
Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and a spectrum of forms in between.
1. Introduction
The paper characterizes how broken Sobolev spaces and interface spaces support DPG formulations, then applies the framework to Fortin operators and Maxwell equations. It derives stability and error-analysis consequences while comparing multiple Maxwell formulations.
- Motivation: Broken Sobolev spaces remove interelement continuity constraints to localize DPG computations.The paper studies this breaking process and its relation to hybridization.
- Interface spaces: Theorem 2.3 characterizes interface spaces connecting broken and unbroken spaces through duality and natural norms.These infinite-dimensional interface spaces provide the precise connection between the two settings.
- Broken formulations: Theorem 3.1 gives sufficient conditions for stability of broken variational forms to follow from stability of unbroken counterparts.This observation simplifies several previous DPG analyses.
- Fortin operators: A commuting sequence of Fortin operators for H1(K), H(curl,K), and H(div,K) satisfies moment conditions and required norm estimates.The operators are constructed on a single tetrahedral element for the Maxwell analysis.
- Maxwell application: DPG Maxwell schemes inherit stability from the exact equations and remain stable on arbitrarily coarse meshes, unlike standard discretizations whose stability may require sufficiently fine meshes.This property is particularly relevant to adaptive meshing.
- Maxwell formulations: Theorem 6.3 shows that stability of one Maxwell formulation implies stability of five others, although their DPG discretizations converge in different norms.The formulations include different choices of strong and weak enforcement.
2. Breaking Sobolev spaces
The section defines broken Sobolev spaces and interface traces on a mesh, then characterizes their norms and their relation to globally conforming spaces. Duality and orthogonality identities recover unbroken spaces from broken functions and interface variables.
- Definitions: The mesh-dependent broken spaces H1(Ω_h), H(curl,Ω_h), and H(div,Ω_h) impose elementwise regularity without interelement continuity.Their easier discretization follows from the absence of continuity requirements at interfaces.
- Trace and interface spaces: Elementwise trace operators generate interface variables, but shared interfaces require single-valued interpretations and careful treatment of trace restrictions.The interface construction addresses multivalued traces and potentially undefined restrictions on interface pieces.
- Interface norms: Interface spaces are equipped with quotient, or minimum-energy-extension, norms defined by minimizing broken-space extension norms.The norms can be interpreted through extensions solving appropriate local boundary-value problems.
- Duality: Theorem 2.3 establishes duality identities for the paired trace spaces associated with H1, H(curl), and H(div).For the H(curl) case, the interface norm equals the norm of a minimum-energy extension and an equivalent Riesz-map representation.
- Recovering unbroken spaces: Orthogonality against interface spaces characterizes conforming subspaces inside broken spaces.For example, a broken scalar function belongs to H1_0(Ω) exactly when its interface pairing vanishes for every appropriate interface function.
F P ˚ Hpcurl, Ωq ðñ x ˆE%, Fyh “ 0 @ ˆE% P H´1{2pdiv, Ωhq. (2.11c)
The section proves the H(curl) characterization of conforming fields through interface orthogonality. Elementwise integration by parts converts vanishing interface pairings into global distributional regularity and boundary conditions.
- Orthogonality characterization: A field F in the broken H(curl) space belongs to the conforming subspace exactly when its pairing with every interface function vanishes.This is the H(curl) equivalence stated in (2.11c).
- Proof mechanism: The proof uses elementwise integration by parts to show that vanishing interface terms make the distributional curl coincide with the piecewise L2 curl.This establishes global H(curl) regularity.
- Boundary condition: The same argument then yields the required homogeneous tangential boundary condition, placing F in the conforming subspace.The boundary trace is recovered after a second integration-by-parts argument.
3. Breaking variational forms
This section develops an abstract framework for replacing unbroken variational formulations with broken ones while preserving stability. The framework characterizes the required spaces and shows how it applies across several elliptic formulations.
- Framework: The section investigates when reformulating a variational problem with broken spaces preserves stability.The abstract result is followed by examples using broken Sobolev spaces and interface spaces.
- Framework: The broken formulation augments the original trial space with an interface space and uses a broken test space.Its form combines the original bilinear form with an interface contribution.
- Abstract stability result: Under Assumptions 1 and 2, stability of the unbroken form implies stability of the corresponding broken form.The proof controls the original and interface unknowns in a triangular sequence.
- Abstract stability result: When the unbroken nullspace is trivial, the broken problem is uniquely solvable and its original solution component agrees with the unbroken solution.The theorem also identifies the relevant nullspaces under the stated interface-space condition.
- Examples: For the primal elliptic example, the broken formulation is wellposed and remains uniquely solvable for a more general right-hand side.The application verifies the abstract assumptions using coercivity and interface-space norm identities.
- Examples: The framework analyzes strong, primal, and ultraweak formulations together, and establishes wellposedness for the resulting family under the stated coefficient assumptions.The formulations differ in which equations are imposed strongly or weakly and in the choice of broken variables and interface traces.
4. The DPG method
This section introduces the DPG method and explains why broken test spaces make its idealized test computations local. It states the Fortin-based conditions yielding solvability, quasi-optimality, and a posteriori error estimates.
- DPG formulation: DPG uses finite-dimensional trial and test subspaces, with approximately optimal tests generated by a discrete trial-to-test operator.The operator is defined through the test-space inner product and variational form.
- DPG formulation: When the test space is broken, computing the trial-to-test operator decomposes into small independent element problems.This localization is the practical reason broken spaces are important for implementation.
- A posteriori estimation: Residual problems in broken test spaces provide element-wise error indicators, with estimates that include data-approximation error.The estimator can be computed for any trial approximation, not only the discrete solution.
- Stability and error analysis: A Fortin operator maps the continuous test space into the discrete test space while preserving the bilinear form on discrete trial functions.The defining condition is b(w_h, v − Πv) = 0 for all discrete trial functions and continuous tests.
- Stability and error analysis: Under the Fortin condition, an inf-sup bound, and a trivial test-space nullspace, the DPG method is uniquely solvable and quasi-optimal.The quasi-optimal estimate bounds the error by the best approximation error in the trial space.
- Fortin operators: The paper notes that the Fortin operator is problem specific, while several useful constructions apply to common broken Sobolev test spaces.The subsequent construction extends this toolkit to H(curl, Ω_h).
5. Fortin operators
This section constructs element-level Fortin operators for the Sobolev sequence H1, H(curl), and H(div) on tetrahedra. Their moment conditions, norm bounds, and commuting structure support DPG error analysis.
- Construction framework: The Fortin-operator construction is performed element by element because the DPG test spaces are broken.The setting is a geometrically conforming tetrahedral finite element mesh.
- Operator properties: On each tetrahedron, the constructed operators map H1, H(curl), and H(div) into polynomial subspaces while satisfying moment conditions and norm estimates.Theorem 5.1 provides the operator existence and stability framework used in the later analysis.
- Construction framework: The operators fit into a commuting exact-sequence diagram linking H1, H(curl), H(div), and L2 with polynomial finite element spaces.The commuting identities connect the continuous differential operators to their discrete counterparts.
- Operator properties: The H1 Fortin operator is uniquely determined by interior and boundary moment conditions and satisfies an H1 norm bound.Its construction separates the boundary mean from the zero-mean trace component.
- DPG application: The resulting Fortin operators supply the approximation tools needed to verify the DPG Fortin assumption and recover established error estimates.The paper applies the operators to the primal DPG formulation after verifying the corresponding inf-sup condition.
6. Maxwell equations
The paper develops and analyzes a DPG method for time-harmonic Maxwell equations in a perfectly conducting cavity, using broken test spaces and interface unknowns. It establishes error estimates and relates the stability of multiple Maxwell formulations.
- Problem setting: The Maxwell model considers harmonic electromagnetic fields in a perfectly conducting cavity with elementwise-constant positive material parameters.The analysis assumes a fixed frequency that is not a cavity resonance.
- Primal DPG formulation: The primal DPG method is obtained by breaking the standard Maxwell variational formulation and integrating by parts element by element.The method uses broken H(curl) test functions and an independent tangential magnetic-field interface unknown.
- Discretization: The numerical scheme discretizes trial and test spaces with conforming H(curl) fields, interface traces, and elementwise Nédélec polynomial test functions.The discrete spaces are defined as subspaces X_h and Y_h of the continuous trial and broken test spaces.
- Stability and error analysis: The resulting DPG convergence estimate requires no sufficiently small mesh size, a property described as absolute stability.The error bound depends on frequency, polynomial degree, and mesh shape regularity through the stated constant.
- Stability and error analysis: The error analysis derives the broken inf-sup condition from the corresponding unbroken formulation and applies the abstract DPG stability framework.The proof verifies the unbroken inf-sup condition first, then uses the developed transfer results and the relevant assumptions.
- Equivalence of formulations: The strong, primal, dual mixed, mixed, and ultraweak Maxwell formulations have equivalent stability statements, with one formulation’s stability implying five others.The equivalence is proved using the closed range theorem and includes the formulations indexed by E, H, S, U, D, and M.
7. Numerical studies
The numerical studies examine DPG performance for smooth and singular Maxwell solutions using primal and ultraweak formulations, uniform and adaptive meshes, and residual-based convergence assessment. The reported smooth-solution experiments show optimal convergence, while the singular Fichera-oven experiments show residual reduction and differing formulation behavior.
- Smooth solution: The smooth-solution primal formulation achieves optimal convergence rates on tetrahedral meshes with the tested discrete spaces.The observed rates are reported for both the H(curl, Ω)-error and residual norm η.
- Smooth solution: Optimal convergence rates are again obtained on cubic meshes using first-type H(curl, Ω)-conforming Nédélec hexahedra.The tensor-product polynomial spaces are revised for the hexahedral meshes.
- Smooth solution: The ultraweak formulation also shows optimal convergence rates for the smooth solution when the adjoint graph norm is used for the test space.Only the interior E and H errors in the L2(Ω)-norm are reported.
- Singular solution: For the singular Fichera-oven problem, adaptive primal iterations drive the computed residual to zero while refining the mesh near the singular geometry.The exact solution is unavailable, so convergence is assessed through the residual estimator η.
- Singular solution: The primal and ultraweak formulations approximate the same solution at different speeds, with the ultraweak formulation appearing faster but requiring more local unknowns and computations.Their interface-unknown counts are identical, whereas the ultraweak formulation has more total unknowns.
8. Conclusion
The paper develops stability and analysis tools for broken DPG formulations, including duality-based interface results and a new Fortin operator for Maxwell discretizations. It also establishes stability equivalence across six Maxwell formulations while documenting practical convergence differences and remaining parameter-dependence limitations.
- Conclusion: Theorem 3.1 simplifies DPG analysis by inheriting stability of broken formulations from known stability of unbroken formulations.The technique applies beyond Maxwell equations to various DPG methods.
- Conclusion: A new Fortin operator provides discrete stability for the Maxwell discretization.The construction is part of the paper’s complete Maxwell DPG analysis.
- Conclusion: Theorem 2.3 uses duality identities to verify the critical interface inf-sup assumption for broken Sobolev spaces.The paper also gives a simple technique for proving the relevant duality identities.
- Conclusion: Theorem 6.3 proves that wellposedness of any one of six Maxwell formulations implies wellposedness of all the others.Numerical experiments nevertheless show practical convergence differences among the formulations.
- Conclusion: The analysis does not provide p-explicit convergence results or track dependence on the wavenumber ω.The authors identify alternative stability analyses and more complex parameter-tracking techniques as open directions.