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A Super-Convergent Quasi-Discontinuous Galerkin Method for Advection-Diffusion Problems

Daniel Y Le Roux

arXiv:2610.29818v1math.APcs.NAmath.NA

TL;DR

The study examines discretizations of the 2D advection-diffusion equation and their stability and dispersion behavior. The paper analyzes linear P1NC and P1DG discretizations stabilized with numerical fluxes through Fourier-derived dispersion relations. The P1NC discrete frequency has O(h4) accuracy, a super-convergent result compared with the P1DG discrete frequency.

  • Problem

    The study examines discretizations of the 2D advection-diffusion equation and their stability and dispersion behavior.

    Source: §1 · p. 1, §2 · p. 2

  • Method

    The paper analyzes linear P1NC and P1DG discretizations stabilized with numerical fluxes through Fourier-derived dispersion relations.

    Source: §5 · p. 14

  • Results

    The P1NC discrete frequency has O(h4) accuracy, a super-convergent result compared with the P1DG discrete frequency.

    Source: §5 · p. 14

  • Takeaways & Limitations

    Within the examined schemes and fluxes, discrete-frequency analysis establishes long-time stability and shows greater frequency accuracy for P1NC than P1DG.

    Source: §5 · p. 14

    The model assumes constant advection speed and diffusion coefficient, with initial data and forcing in L2(Ω) and sufficiently regular boundary data.

    Source: §2 · p. 2

Abstract

from arXiv · show

The aim of the present study is to develop discontinuous Galerkin and nonconforming finite-element methods for solving the 2D advection-diffusion equation. The discrete formulations are stabilized by means of a family of numerical fluxes and the dispersion relations are constructed via a Fourier/stability analysis. The linear discontinuous P1DG and non conforming P1NC Galerkin discretizations are studied. A long time stability result is proven for the P1NC and P1DG schemes and fluxes examined here by inspecting the discrete frequencies, solutions of the dispersion relations. A super-convergent result is also demonstrated for the P1NC discrete frequency (O(h 4 ) accuracy) compared to the P1DG one (O(h 2 ) accuracy). These results are in line with those obtained in a previous paper for the shallow-water system.

1. Introduction.

The paper studies linear P1 nonconforming and discontinuous Galerkin discretizations for the 2D advection-diffusion equation, extending a super-convergence finding previously demonstrated for a shallow-water system. It analyzes their dispersion relations and reports long-time stability and higher discrete-frequency accuracy for P1NC than P1DG.

1. Introduction.

  • DG methods approximate discrete solutions with discontinuous, usually piecewise-polynomial basis functions.The introduction traces their use from transport equations to hyperbolic equations and broader computational applications.Source: §1 · p. 1
  • For higher-order spatial derivatives, LDG rewrites the problem using first-order derivatives and selects numerical fluxes to ensure stability.The cited introduction describes LDG as retaining DG properties including stability, high-order accuracy, and flexibility for h-p adaptivity and complex geometries.Source: §1 · p. 1
  • The study develops and compares linear P1NC and P1DG Galerkin discretizations for the 2D advection-diffusion equation.The motivation follows a super-convergence result for discrete frequencies in a shallow-water system using P1NC.Source: §1 · p. 1
  • The dispersion analysis constructs relations for the discretizations and assesses long-time stability through their discrete frequencies.The formulations use a family of numerical fluxes.Source: §1 · p. 2
  • In the comparison, the P1NC discrete frequency has O(h^4) accuracy, whereas the P1DG discrete frequency has O(h^2) accuracy.Source: §1 · p. 2

2. The problem model.

The model is a 2D advection-diffusion equation on a Lipschitz domain, with Dirichlet and Neumann boundary portions. Its stated setting uses constant advection speed and diffusion coefficient, with initial data and forcing in L2(Ω) and sufficiently regular boundary data.

2. The problem model.

  • The domain is Lipschitz, and its boundary is split into disjoint Dirichlet and Neumann portions.The boundary condition on ΓN prescribes the normal derivative using θN.Source: §2 · p. 2
  • The governing equation combines time evolution, advection, diffusion, and a source term on Ω × (0,T).The model specifies Dirichlet data on ΓD and a Neumann condition on ΓN.Source: §2 · p. 2
  • The initial solution and forcing are chosen in L2(Ω), while the prescribed boundary functions are assumed sufficiently regular.The regularity condition applies to uD on ΓD × (0,T) and θN on ΓN × (0,T).Source: §2 · p. 2
  • The advection speed is a constant vector and the diffusion coefficient ν is constant.The model assumes a1 > 0 and a2 > 0 without loss of generality.Source: §2 · p. 2

3. Discretization of the model equations.

The paper derives discrete weak formulations for P1NC and P1DG using elementwise polynomial spaces and numerical traces on inter-element boundaries. It then specifies each discretization and the Fourier-analysis setting used to study their dispersion relations.

3.1. The discrete weak formulation.

  • The approximate solutions are obtained from elementwise weak formulations that replace discontinuous fields on interior edges with numerical traces.The traces are chosen to make the method consistent and stable, using interface means and jumps; the classical upwind flux is used for the advective trace.Source: §3.1 · pp. 3–4
  • The LDG formulation introduces an auxiliary flux variable θ, which can be solved locally in terms of uh and eliminated when ξ = 0.The flux parameters are chosen to satisfy homogeneity constraints, and the text reports that ξ provides no particular accuracy or stability information in the analyzed dispersion relations.Source: §3.1 · p. 4
  • Fourier analysis uses a uniform mesh of biased right triangles, periodic boundaries, and zero forcing.Symmetry and Fourier analysis reduce the computation to selected nodal equations for both schemes.Source: §3.1 · p. 5

3.2. The P NC

  • P1NC unknowns lie at edge midpoints shared by adjacent triangles, so the basis is continuous there but discontinuous elsewhere along triangle boundaries.Its orthogonal basis gives a diagonal mass matrix, and the local NC method computes θh from uh alone.Source: §3.2 · pp. 6, 8

3.3. The P DG

  • P1DG unknowns belong exclusively to individual elements, making the approximation discontinuous between triangles.The study adopts the mid-edge basis option because it gives diagonal mass matrices; vertex and mid-edge choices yield the same dispersion relation.Source: §3.3 · p. 8

4. The Fourier / dispersion analysis.

The analysis uses Fourier modes to derive discrete dispersion relations for the P1NC and P1DG schemes and compare their frequencies with the continuous counterpart. The asymptotic results show superconvergence for P1NC relative to P1DG and examine stability under specified parameter settings.

4.1. The generalized eigenvalue system.

  • The analysis derives a generalized eigenvalue problem by substituting Fourier expansions into selected P1NC and P1DG discrete equations.The resulting problem is 3n × 3n; different face or edge midpoint types must be considered because the schemes’ unknowns occupy multiple node types.Source: §4.1 · pp. 10–11

4.2. The dispersion relations.

  • The dispersion relation is a degree-n polynomial in frequency whose coefficients depend on wave numbers, mesh spacing, flux parameter, diffusion, and advection parameters.Its roots are compared with the continuous frequency, with asymptotic expansions computed separately for cases where general symbolic expansions are unavailable.Source: §4.2 · p. 12
  • The P1NC discrete frequency is reported to be super-convergent relative to P1DG, with asymptotic expansions examined for separate parameter cases.The analysis treats the cases without advection and without diffusion separately; the latter also sets p = 0.Source: §4.2 · p. 12
  • The identified high-frequency modes decay exponentially fast when p + pj > 0 and are stated to have no practical impact.For the zero-diffusion, p = 0 case, the reported frequencies are also stable and decay exponentially fast.Source: §4.2 · p. 13
  • The schemes’ discrete frequencies are stable at leading order under the parameter conditions reported for P1NC and P1DG.The supplied passage gives thresholds p > −2 and p > −4, respectively, and identifies a stable parameter choice in the following theorem.Source: §4.2 · p. 13
  • For P1DG, the parameter p ≃ −1.3822831 is obtained by solving a polynomial, while its other four roots are reported to yield unstable results.The expansion for this parameter includes a frequency-error term proportional to νk^6h^4.Source: §4.2 · p. 13
  • The normalized-frequency plots cover 0 ≤ kh, lh ≤ π/10 and report maximum and minimum values for the indicated schemes and parameter choices.The text says these results illustrate accurate P1NC behavior at least for small wave numbers.Source: §4.2 · pp. 13–14

5. Conclusion.

The study examines Fourier/stability analysis for linear P1DG and P1NC discretizations of the 2D advection-diffusion equation. It reports long-time stability for the examined schemes and fluxes, and higher discrete-frequency accuracy for P1NC than P1DG.

5. Conclusion.

  • The paper presents Fourier/stability analysis of the 2D advection-diffusion equation for linear P1DG and P1NC discretizations as a first study.Source: §5 · p. 14
  • Long-time stability is proven for the examined P1NC and P1DG schemes and numerical fluxes by inspecting discrete frequencies.Source: §5 · p. 14
  • P1NC discrete frequency has O(h^4) accuracy, compared with O(h^2) for P1DG.Source: §5 · p. 14
  • The reported results are in line with those obtained for the shallow-water equations [12].Source: §5 · p. 14

Appendix A. The vectors Λ1, Λ2, Λ3, Λ4, for the P NC

Appendix A presents algebraic entries for the P1NC vectors, including expressions in indexed u terms and θx and θy terms.

Appendix A.

  • The listed P1NC vector entries include algebraic expressions in indexed u terms.Source: Appendix A · pp. 14–15
  • The passage gives an algebraic expression involving u-variables, namely −4u1 + u2 + u3 + u4 − u5 − u10 + u11 + u14 + u15 − u16 + u17 − u18 + u19.Source: Appendix A · p. 15
  • The displayed P1NC vector expressions include terms involving θx and θy.Source: Appendix A · p. 15

Appendix B. The vectors Λ1, Λ2, Λ3, Λ4, for the P DG

Appendix B presents algebraic entries for the P1DG vectors, including expressions in indexed u terms and θx and θy terms.

Appendix B.

  • The listed P1DG vector entries include algebraic expressions in indexed u terms.Source: Appendix B · p. 15
  • The cited passages display several expressions: linear combinations of u variables in s12.p1–p3 and an expression involving θx and θy in s12.p4, without labeling them as P1DG vectors.Source: Appendix B · p. 15
  • The displayed P1DG vector expressions include terms involving θx and θy.Source: Appendix B · p. 15

Appendix D. The matrices A1, A2, A3, for the P DG

Appendix D displays matrix-form material for the P1DG matrices A1, A2, and A3.

Appendix D.

  • The appendix displays bracketed matrix-form material for the P1DG matrices A1, A2, and A3.Source: Appendix D · p. 16
  • The appendix’s matrix material is identified as belonging to the P1DG formulation.Source: Appendix D · p. 16
  • The displayed matrix entries include α and β terms.Source: Appendix D · p. 16
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