A discontinuous Galerkin method for elliptic-hyperbolic equations

1. Problem Statement

The paper addresses the numerical solution of a class of second-order linear mixed-type partial differential equations (PDEs), specifically the Frankl operator:
$$Lu := K(y)u_{xx} + u_{yy} = f$$
where the coefficient $K(y)$ changes sign within the computational domain $\Omega \subset \mathbb{R}^2$.

• Elliptic Region: $y > 0$ (where $K(y) > 0$).
• Hyperbolic Region: $y < 0$ (where $K(y) < 0$).
• Parabolic Line: $y = 0$ (where $K(y) = 0$).

The specific focus is on the Tricomi problem, a classic open boundary value problem where Dirichlet conditions are imposed on the elliptic boundary ($\Gamma_0$) and one characteristic boundary ($\Gamma_1$), while the other characteristic boundary ($\Gamma_2$) is left free. This problem arises in transonic gas flow (transition between subsonic and supersonic regimes) and other physical phenomena.

2. Methodology: Discontinuous Galerkin (DG) Formulation

The authors propose a Discontinuous Galerkin (DG) method that directly discretizes the second-order equation without converting it to a first-order system. The formulation is inspired by the continuous energy-integral method used to prove well-posedness.

Key Components of the Method:

1. Morawetz Multipliers:
   The core of the formulation relies on multiplying the PDE by a Morawetz multiplier $Mv = b v_x + c v_y$, where $b$ and $c$ are specific scalar functions satisfying positivity and boundary conditions (Assumptions A1–A4). This technique, derived from the Friedrichs $abc$-method, allows the derivation of energy estimates essential for stability.

2. Variational Formulation:
   The weak form is constructed by integrating by parts over each element $T \in \mathcal{T}_h$. The formulation includes:

   • Volume terms: Involving the operator $L$ and the multiplier.
   • Flux terms: Handling jumps across interior facets using averages $\{\{\cdot\}\}$ and jumps $[\![\cdot]\!]$.
   • Stabilization terms: To ensure coercivity in a mesh-dependent norm, the authors add penalty terms for:
     • Jumps in the solution $[\![u]\!]$ (scaled by $\gamma_1/h_F^3$).
     • Jumps in the first derivatives $[\![u_x]\!]$ and $[\![u_y]\!]$ (scaled by $\gamma_2 p^2/h_F$).
     • Tangential derivatives on Dirichlet boundaries (scaled by $\gamma_3 p^2/h_F$).

3. Discrete Spaces:
   The method is analyzed for three types of discrete spaces $V_h$:

   • Standard Polynomials: Piecewise polynomials of degree $p$ ($P_p(\mathcal{T}_h)$).
   • Quasi-Trefftz Spaces: Subspaces where basis functions satisfy the PDE approximately (matching Taylor expansions of the exact solution up to order $p-2$).
   • Embedded Trefftz Spaces: Subspaces where the PDE residual is projected to zero in a weak sense (using $L^2$ projection), avoiding explicit construction of Trefftz bases.

3. Key Contributions & Theoretical Results

A. Well-Posedness and Stability

• Coercivity: The authors prove that the discrete bilinear form is coercive in an energy norm $|||\cdot|||$, provided the penalty parameters $\gamma_2$ and $\gamma_3$ are sufficiently large (specifically $\gamma_{2,3} \geq \gamma^*$). This is achieved via the Morawetz multiplier technique, which yields a positive definite energy estimate.
• Boundedness: The bilinear form is shown to be bounded in a stronger "residual" norm $|||\cdot|||_L$, which includes the $L^2$ norm of the residual $Lu$.
• Existence/Uniqueness: By establishing coercivity and boundedness, the existence and uniqueness of the discrete solution are guaranteed for general polygonal meshes and arbitrary polynomial spaces.

B. Error Analysis and Convergence Rates

The paper derives a priori error estimates in the energy norm:

• Standard Polynomials: The method achieves convergence rates of order $O(h^{p-1})$ in the energy norm for polynomial degree $p \geq 2$.
  • Note: The convergence is suboptimal by one order in $h$ compared to standard elliptic problems (which typically achieve $O(h^p)$). This is a known characteristic of mixed-type problems and is consistent with previous finite element analyses.
• Quasi-Trefftz Spaces: The method achieves $O(h^{p-1})$ convergence in the energy norm.
  • Advantage: The dimension of the quasi-Trefftz space is $O(p)$ (linear in $p$), compared to $O(p^2)$ for standard polynomials, leading to a significant reduction in degrees of freedom (DoFs).
• Embedded Trefftz Spaces: Numerical experiments suggest similar convergence rates to the standard and quasi-Trefftz cases, with the added benefit of not requiring explicit Taylor expansions of coefficients.

C. Least-Squares Variant

The authors also investigate a least-squares variant of the DG formulation. While theoretically consistent, numerical results indicate it does not significantly improve accuracy for Trefftz-based spaces (which already have small residuals) and may slightly degrade accuracy for standard polynomials.

4. Numerical Results

The theoretical findings are validated through numerical experiments on the Tricomi equation ($K(y)=y$):

• $h$-convergence: Confirmed $O(h^{p-1})$ rates in the energy norm for all three spaces. $L^2$ errors showed rates of $O(h^p)$ or $O(h^{p+1})$ depending on parity of $p$.
• $p$-convergence: On a fixed mesh, increasing $p$ showed:
  • Standard Polynomials: Root-exponential convergence $O(e^{-B\sqrt{N_{dofs}}})$.
  • Trefftz Methods: True exponential convergence $O(e^{-A N_{dofs}})$, achieving higher accuracy for the same number of DoFs.
• Sensitivity: The method is robust with respect to penalty parameters. While very small parameters lead to instability, the method remains stable and accurate over a wide range of values, with embedded Trefftz spaces showing the highest robustness.

5. Significance and Impact

• Unified Framework: This work provides a unified DG framework for mixed-type equations, directly handling the second-order form without the complexity of first-order systems.
• Efficiency: The integration of Trefftz and quasi-Trefftz methods into the DG setting for mixed-type problems is a novel contribution. It demonstrates that these specialized spaces can drastically reduce the computational cost (DoFs) while maintaining optimal convergence rates for the specific physics of the problem.
• Robustness: The method handles the transition from elliptic to hyperbolic regions and the parabolic degeneracy naturally, without requiring special mesh alignment or domain decomposition strategies.
• Practical Application: The use of Morawetz multipliers in the discrete setting bridges the gap between continuous well-posedness theory and numerical implementation, offering a stable and convergent scheme for challenging transonic flow problems.

In summary, the paper presents a theoretically sound and numerically efficient DG method for elliptic-hyperbolic equations, leveraging Morawetz multipliers for stability and Trefftz spaces for computational efficiency.