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An analysis of the practical DPG method
Jay Gopalakrishnan, Weifeng Qiu
TL;DR
The paper addresses the gap between ideal DPG theory and the fully implementable practical method, whose trial-to-test operator must be approximated. It develops a complete error analysis and shows that polynomial test enrichment preserves optimal convergence for Laplace problems, with analogous results for linear elasticity and conditioning estimates.
Problem
Ideal DPG stability relies on a trial-to-test operator whose application is generally infeasible because it requires solving an infinite-dimensional problem.
Method
The paper analyzes practical DPG methods that approximate the trial-to-test operator using finite-dimensional polynomial test spaces and establishes stability through Fortin-operator conditions.
Results
For the Laplace equation, optimal convergence is ensured when the test-polynomial degree satisfies r ≥ p + N; similar error results are established for linear elasticity.
Takeaways & Limitations
The practical DPG method retains the desired convergence behavior while making the trial-to-test computation implementable, and its stiffness matrix has comparable conditioning to other methods.
Takeaways & Limitations
The stated elasticity result assumes an element-wise constant compliance tensor, while the general analysis relies on explicit stability and projection assumptions.
Abstract
from arXiv · showhide
In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $p$ on each mesh element. Earlier works showed that there is a "trial-to-test" operator $T$, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator $T$ is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply $T$. In practical computations, $T$ is approximated using polynomials of some degree $r > p$ on each mesh element. We show that this approximation maintains optimal convergence rates, provided that $r\ge p+N$, where $N$ is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.
1. Introduction
The paper completes the error analysis of the practical DPG method, replacing the infeasible ideal trial-to-test operator with a finite-dimensional approximation while retaining implementability.
- 1. Introduction: The paper extends prior ideal-DPG analysis to fully implementable practical methods for the Laplace equation and linear elasticity.The analysis is presented as applicable more generally to other equations as well.
- 1. Introduction: The ideal DPG method is stable, but applying T is generally infeasible because it requires solving an infinite-dimensional problem.Closed-form application is available only for certain special problems.
- 1. Introduction: The practical DPG method approximates the ideal trial-to-test operator T in a finite-dimensional test space V_r, making the method implementable.T_r is defined through the same inner-product relation as T, but with test functions restricted to V_r.
- 1. Introduction: Applying T_r requires Gram-matrix inversion, which is prohibitively expensive in standard formulations when the test space is H1(Ω).For H1(Ω), the inversion is described as expensive as solving the Laplace equation.
- 1. Introduction: Hybridization makes the practical operator local by using discontinuous test functions, yielding a block-diagonal Gram matrix with one block per mesh element.The resulting application of T_r reduces to block-diagonal inversion.
2. The approach to analysis
The analysis establishes practical-DPG stability through a discrete inf-sup condition and a Fortin operator, then derives consequences for test bases and stiffness-matrix conditioning.
- 2. The approach to analysis: A Fortin operator into the standard polynomial test space V_r is sufficient for convergence of the practical DPG method.This provides a route to analyze practical methods without working directly with the ideal test space.
- 2. The approach to analysis: Theorem 2.1 derives discrete stability from a discrete inf-sup condition under the stated injectivity, inf-sup, continuity, and projection assumptions.The proof applies Babuška’s theory and proceeds through three inf-sup steps.
- 2. The approach to analysis: The proof compares suprema over the full and discrete test spaces, using the projection assumption and the identity involving T_r to establish the required inf-sup condition.The argument concludes by showing the relevant suprema coincide.
- 2. The approach to analysis: Injectivity of T_r implies that the computed test functions form a basis with the same dimension as the trial space.Applying T_r to any basis of U_h produces a basis for the practical test space.
- 2. The approach to analysis: The resulting stiffness matrix is symmetric positive definite, with condition numbers comparable to those of other methods under the analysis assumptions.The conditioning argument uses the inf-sup and continuity bounds.
3. First example: Laplace equation
For the Laplace equation, the practical DPG method uses polynomial test spaces of degree r≥p+N and retains the ideal method’s convergence guarantees. The analysis constructs the required projections, establishes convergence rates, and discusses reduced enrichment and conditioning.
- The Laplace formulation uses an ultraweak DPG setting with trial variables for flux, solution, traces, and normal fluxes, paired with broken test spaces.The bilinear form is defined elementwise over H(div, Ω_h) and H1(Ω_h), while the solution component satisfies the Laplace equation with homogeneous Dirichlet data.
- A Fortin projection into the polynomial test space verifies the stability assumptions needed for practical DPG convergence.The construction uses polynomial projections and Piola mappings on simplices, with r=p+N providing the dimension relations required by the local systems.
- Choosing r≥p+N for the discrete test space yields the practical DPG convergence result for the Laplace equation.The test space uses elementwise polynomials and therefore permits local application of the approximated trial-to-test operator.
- Different enrichment degrees can be used for scalar and flux test components, so assigning the same degree to every component is unnecessary.The revised test space is smaller when N≥3 while preserving the same convergence result.
- The resulting h-convergence estimates hold for N=2 or 3 over the stated regularity range 1/2<s≤p+1.These rates follow from the practical method’s error estimate under the assumptions of Theorem 3.4.
- On quasiuniform tetrahedral meshes, the DPG stiffness matrix has a spectral condition number no worse than standard methods.The conditioning result is stated for local basis functions mapped from the reference simplex.
4. Second example: Linear elasticity
The paper analyzes a practical DPG method for linear elasticity by replacing the ideal test construction with polynomial test functions. Under r ≥ p + N and element-wise constant compliance, it proves a quasioptimal error estimate and analogous results to the Laplace analysis.
- Scope and robustness: The practical method is analyzed for N = 2 or 3 and inherits robustness with respect to the Poisson ratio from the corresponding ideal elasticity method.The cited ideal method is described as locking-free for homogeneous isotropic materials.
- Formulation: The linear-elasticity formulation uses symmetric H(div) stress fields, H1 displacement fields, skew-symmetric multiplier fields, and an additional scalar component.The compliance tensor is assumed element-wise constant, and the formulation recovers the elasticity equations with homogeneous displacement boundary conditions.
- Discrete spaces: The method uses degree-p polynomial trial fields and degree-r polynomial test fields, with the discrete test space obtained through a localized trial-to-test construction.The trial and test spaces are defined elementwise, and the practical method specifies the enriched test degree r.
- Verification: The proof verifies the abstract assumptions through projection operators, polynomial moment conditions, commutativity properties, and mapped constructions on reference simplices.These ingredients establish the operator properties needed to invoke the abstract error theorem.
- Main result: Under r ≥ p + N and element-wise constant compliance, the practical linear-elasticity DPG solution satisfies the theorem’s error estimate.The proof applies the abstract quasioptimality theorem after verifying its assumptions.