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A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs

Vijay Kumar, Gautam Singh

arXiv:2610.35343v1math.NAcs.NA

TL;DR

Small perturbation parameters can create sharp layers that standard discretizations may not resolve, motivating methods suited to these solutions. The paper combines ADI splitting with DDG on a Shishkin mesh and analyzes the resulting scheme. It reports coercivity and stability results and an error estimate for the discretization.

  • Problem

    Small perturbation parameters create sharp boundary or interior layers that standard methods may fail to resolve, making accurate numerical treatment difficult.

    Source: §1 · p. 2

  • Method

    The method splits a two-dimensional problem into one-dimensional subproblems, applies DDG on a piecewise-uniform Shishkin mesh, and uses backward Euler in time.

    Source: §1 · p. 4, §4 · p. 9, §7 · p. 27

  • Results

    The analysis establishes coercivity and stability properties and provides a discretization error estimate for the numerical solutions.

    Source: §4.3 · p. 14, §5.2 · p. 23

  • Takeaways & Limitations

    The directional splitting reduces degrees of freedom and computational cost and memory requirements relative to direct two-dimensional discretization.

    Source: §1 · p. 4, §7 · p. 27

    The stated model assumes positive directional advection, nonnegative reaction components, and sufficiently smooth source and initial data.

    Source: §2 · p. 5

Abstract

from arXiv · show

This study proposes two-dimensional one-parameter singularly perturbed parabolic problems and develops an operator-splitting direct discontinuous Galerkin (DDG) method for their numerical solution. To improve computational efficiency, an alternating direction implicit (ADI) splitting approach decomposes the original two-dimensional problem into two separate one-dimensional subproblems. The spatial discretization is performed using the DDG method on a piecewise-uniform Shishkin mesh, enabling accurate resolution of the sharp boundary layers that arise in singularly perturbed problems (SPPs). For the temporal discretization is carried out with the backward Euler scheme to ensure robustness and stability. The theoretical analysis establishes an optimal convergence rate of order $k$ in the spatial discretization, where $k$ denotes the degree of the polynomial basis functions, and $(\Delta t)$-order convergence in temporal dimensions. We prove that the resulting fully discrete scheme is consistent, stable, and convergent. Numerical experiments confirm the theoretical results and demonstrate the effectiveness and robustness of the proposed method.

1. Introduction

Singularly perturbed problems develop sharp layers that can challenge standard discretizations, motivating stable, accurate methods on layer-adapted meshes. This paper addresses the uninvestigated combination of ADI operator splitting and DDG for two-dimensional one-parameter parabolic problems.

1. Introduction

  • Small parameters associated with the highest-order derivatives produce sharp spatial variations in boundary or interior layers, making singularly perturbed problems analytically and computationally difficult.The paper notes applications including control theory, transport and flow modeling, and fluid mechanics.Source: §1 · p. 2
  • Standard FDM and FEM may fail to resolve steep boundary or interior layers when the perturbation parameter is small, and conventional discretizations may also produce non-physical oscillations in convection-dominated regimes.Layer-adapted meshes place points strategically near sharp variations; Shishkin meshes use fine and coarse regions, while Bakhvalov meshes use smooth grading.Source: §1 · p. 2
  • The paper develops an ADI-type operator-splitting DDG method for two-dimensional one-parameter singularly perturbed parabolic problems, a combination not systematically investigated previously.The splitting converts the problem into one-dimensional subproblems, reducing degrees of freedom and algebraic-system size.Source: §1 · p. 4
  • The introduction states that the generic positive constant in the analysis is independent of perturbation parameters µ and mesh parameters N and M.Its value may vary between lines.Source: §1 · p. 4

2. The continuous problem

The model is a two-dimensional one-parameter singularly perturbed parabolic initial-boundary value problem, with small diffusion and boundary layers near two edges. Its directional operator splitting advances the solution through successive one-dimensional x- and y-subproblems.

2. The continuous problem

  • The model has small diffusion parameter µ, positive directional advection components, nonnegative reaction components, and sufficiently smooth source and initial data; small µ produces layers near x = 1 and y = 1.The problem is posed on the unit-square spatial domain over (0, T].Source: §2 · p. 5

2.1. ADI operator splitting and problem decomposition

  • The two-dimensional model is decomposed into separate x- and y-direction one-dimensional subproblems and uses backward Euler time discretization.The source is split as r = r1 + r2 subject to boundary compatibility conditions.Source: §2.1 · pp. 5–6
  • For each time interval, the x-subproblem produces an intermediate solution that initializes the y-subproblem, whose result advances the solution to the next time level.The x solve is performed for fixed y; the y solve is performed for fixed x.Source: §2.1 · p. 6

3. A priori bounds and domain discretization

The analysis separates smooth and layer components to derive solution bounds, then constructs layer-adapted spatial meshes and the notation and polynomial spaces used by the DDG discretization.

3.1. A priori bounds for the continuous solution

  • The exact solution is decomposed into smooth and singular components, and the directional solutions are likewise split into smooth and layer components with derivative bounds through the stated orders.The bounds assume sufficient smoothness of the source and directional advection data.Source: §3.1 · p. 7

3.2. Spatial domain discretization

  • A tensor-product, piecewise-uniform Shishkin mesh is used in both spatial directions, with fine and coarse regions separated by transition points near the boundary layers.Each region is uniformly partitioned, with N/2 intervals per subinterval.Source: §3.2 · pp. 7–8

3.3. Preliminaries and notation for the discontinuous Galerkin method

  • The x- and y-direction finite element spaces use polynomials of degree at most k on each mesh element.The mesh partitions combine into a tensor-product partition of the spatial domain.Source: §3.3 · pp. 8–9
  • The DDG notation defines directional jump and average operators at interior nodes and boundary points.The y-direction definitions are analogous to those in the x-direction.Source: §3.3 · p. 9

4. Discontinuous Galerkin approximation

The fully discrete method combines DDG spatial discretization with backward Euler time stepping, using interface corrections and numerical fluxes. The analysis states admissibility conditions for the fluxes and establishes coercivity and stability properties.

4.1. Fully-discrete formulation

  • The fully discrete DDG scheme applies the spatial method to the split subproblems at each time step, with backward Euler time discretization.Interface correction terms are identified as essential for optimal accuracy with higher-order polynomial approximations.Source: §4 · p. 9, §4.1 · pp. 10–11

4.2. Admissibility of the DDG method

  • The numerical fluxes are chosen to ensure stability, and admissibility is characterized by conditions involving constants γ1 and γ2.The text reports admissibility results for the x- and y-direction fluxes and refers to an analytical estimate for η0(k).Source: §4.2 · pp. 12–13, Table 1 · p. 13

4.3. Coercivity and stability analysis of the DDG method

  • Under the flux admissibility conditions, the coercivity result provides a constant independent of the perturbation parameter and mesh size.The paper uses the DG norm, which incorporates the solution and jumps across element interfaces and is slightly stronger than the energy norm.Source: §4.3 · p. 14
  • The stability analysis gives a bound for the numerical solution at each time step, obtained by combining the directional estimates.The stated argument adds the estimates for the two directions and sums over spatial indices.Source: §4.3 · pp. 14–15

5. Error analysis

Section 5 builds the error analysis by separating directional, interpolation, and discretization errors, then combines component estimates for the full numerical solution.

5.1. Interpolation error

  • Each directional error is decomposed into interpolation and discretization parts, whose bounds support the overall error analysis.The interpolation analysis assumes the exact directional solutions lie in H0^1 and H^(k+1).Source: §5 · p. 16, §5.1 · pp. 17–18, §5.2 · pp. 18, 20

5.2. Discretization error

  • The discretization-error analysis establishes bounds for the x- and y-direction subproblems under H^(k+1) regularity.The theorem statements specify the directional solution regularity and numerical solutions, but the supplied passages do not show the bound formulas.Source: §5.2 · pp. 18, 20, 21, 23
  • The full-solution error estimate combines the estimates for the two directional subproblems.The exact solution is defined as u = v+w, with a corresponding combined numerical solution.Source: §5.2 · p. 23

6. Numerical experiments

Two examples with constant and variable coefficients are used to assess errors, convergence rates, and solution profiles. The experiments use polynomial degrees k=1 and k=2 with different time-step scalings.

6. Numerical experiments

  • The two tests cover constant and variable convection coefficients, with the variable terms 1+x(1−x) and 1+y(1−y).Both examples construct the source term and initial condition from the same exact solution.Source: §6 · p. 24
  • For k=1, the time step is Δt=N^-1; for k=2, it is Δt=N^-2.These choices are described as providing sufficient temporal resolution relative to the spatial discretization.Source: §6 · p. 24
  • The source decomposition defines r2 by linear interpolation in y between r(x,0,t) and r(x,1,t).Source: §6 · p. 24
  • Tables report DG-norm errors and observed convergence rates for both examples across μ, with k=1 using Δt=N^-1 and k=2 using Δt=N^-2.The tables cover various values of the singular perturbation parameter.Source: §6 · p. 25, Table 2 · p. 25, Table 3 · p. 26
  • For Example 1, Figure 1 compares exact and DDG solution surfaces at T=1 on a 32 × 32 mesh with Δt=1/32 and μ=10^-4.The plots are presented to demonstrate the method's capability in resolving boundary layers.Source: §6 · p. 25, Figure 1 · p. 26

7. Conclusion

The paper develops an ADI-type operator-splitting DDG method for two-dimensional singularly perturbed parabolic problems. Its analysis establishes spatial order k and first-order temporal convergence, and experiments with constant and variable convection coefficients confirm the error estimates.

7. Conclusion

  • The analysis yields optimal spatial convergence of order k and first-order convergence in time.Source: §7 · p. 27
  • The fully discrete scheme is proved consistent, stable, and convergent, with DG-norm spatial error bounds independent of μ.Source: §7 · p. 27
  • ADI-type splitting decomposes the two-dimensional problem into one-dimensional subproblems, reducing degrees of freedom, computational cost, and memory requirements relative to direct two-dimensional discretization.Source: §7 · p. 27
  • The method uses DDG spatial discretization on a piecewise-uniform Shishkin mesh to resolve boundary layers and backward Euler time integration for unconditional stability.Source: §7 · p. 27
  • The method addresses the lack of systematic study of DDG for two-dimensional singularly perturbed parabolic problems; experiments cover constant and variable convection coefficients.Source: §7 · p. 27
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