[Paper Review] Five lectures on DPG methods

Jay Gopalakrishnan|arXiv (Cornell University)|Jun 3, 2013

Advanced Numerical Methods in Computational Mathematics9 references20 citations

TL;DR

This paper introduces the Discontinuous Petrov-Galerkin (DPG) method as a stable, least-squares-like Galerkin method using optimal test functions derived from a Riesz representation in a nonstandard norm. It proves that with optimal test spaces, the discrete inf-sup condition is preserved, ensuring quasioptimal convergence and robustness, and demonstrates the method's effectiveness through adaptive refinement and numerical experiments on elliptic problems.

ABSTRACT

This is a set of lecture notes introducing graduate students to the topic of Discontinuous Petrov-Galerkin (DPG) methods.

Motivation & Objective

• To present a systematic introduction to Discontinuous Petrov-Galerkin (DPG) methods as a stable class of discontinuous Galerkin methods.
• To establish the theoretical foundation of DPG using optimal test functions derived from the trial-to-test operator.
• To prove that the discrete inf-sup condition is preserved when using optimal test spaces, ensuring well-posedness and convergence.
• To demonstrate the practical utility of DPG through adaptive finite element methods and error estimation.
• To bridge theoretical stability with computational implementation using FEniCS.

Proposed method

• Define the trial-to-test operator $ T: X \to Y $ via $ (Tz, y)_Y = b(z, y) $ for all $ y \in Y $, ensuring optimal test functions.
• Construct the optimal test space $ Y_h^{\text{opt}} = T(X_h) $, where $ X_h $ is the trial space.
• Prove that the discrete inf-sup condition holds with constant $ C_3 = C_1 $ when using $ Y_h^{\text{opt}} $, provided the continuous inf-sup condition holds.
• Apply Theorem 2 to establish quasioptimal convergence: $ \|x - x_h\|_X \leq \frac{C_2}{C_1} \inf_{z_h \in X_h} \|x - z_h\|_X $.
• Implement the DPG method in FEniCS for the Poisson equation and use the residual-based error estimator for adaptive mesh refinement.
• Use a marking strategy based on error indicators to refine elements in the top half of the error distribution, driving convergence toward the solution's layers.

Experimental results

Research questions

• RQ1 Can the discrete inf-sup condition be guaranteed for any finite-dimensional trial space by choosing an appropriate test space?
• RQ2 Does the use of optimal test functions derived from the Riesz representation preserve the stability of the continuous problem in the discrete setting?
• RQ3 Can the DPG method achieve quasioptimal convergence rates without first-order reformulation?
• RQ4 How can a reliable and efficient error estimator be constructed for adaptive DPG methods?
• RQ5 Can adaptive refinement guided by the DPG error estimator effectively resolve sharp layers in the solution, such as those near singularities?

Key findings

• The discrete inf-sup condition (4) holds with $ C_3 = C_1 $ when the test space is chosen as the optimal test space $ Y_h^{\text{opt}} $, ensuring well-posedness of the discrete problem.
• The ideal DPG method using $ Y_h^{\text{opt}} $ achieves quasioptimal convergence with the bound $ \|x - x_h\|_X \leq \frac{C_2}{C_1} \inf_{z_h \in X_h} \|x - z_h\|_X $, where $ C_1 $ and $ C_2 $ are the continuous inf-sup and continuity constants.
• The optimal test function $ Tz $ maximizes the ratio $ \frac{|b(z,y)|}{\|y\|_Y} $, confirming its optimality in the Hilbert space setting.
• Adaptive DPG with the residual error estimator successfully resolves the solution's sharp layers near the origin in the Poisson problem with a highly localized source.
• The FEniCS implementation of the DPG method and adaptive algorithm reproduces the expected convergence behavior and automatically refines regions of high error.
• The method is robust and stable without requiring first-order system reformulation, as shown in the primal DPG formulation.