A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs
Abstract
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 (Δt)-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.
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BibTeX
@misc{kumar2026robust,
title = {{A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs}},
author = {Vijay Kumar and Gautam Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35343},
eprint = {2610.35343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35343},
}
RIS
TY - UNPB
TI - A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs
AU - Vijay Kumar
AU - Gautam Singh
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35343
UR - https://arxiv.org/abs/2610.35343
AB - 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 (Δt)-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.
ER -