Optimal error estimates for a discontinuous Galerkin method on curved boundaries with polygonal meshes
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| Format: | Preprint |
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2026
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| _version_ | 1866914257566695424 |
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| author | Araújo, Adérito Santos, Milene |
| author_facet | Araújo, Adérito Santos, Milene |
| contents | We consider a discontinuous Galerkin method for the numerical solution of boundary value problems in two-dimensional domains with curved boundaries. A key challenge in this setting is the potential loss of convergence order due to approximating the physical domain by a polygonal mesh. Unless boundary conditions can be accurately transferred from the true boundary to the computational one, such geometric approximation errors generally lead to suboptimal convergence. To overcome this limitation, a higher-order strategy based on polynomial reconstruction of boundary data was introduced for classical finite element methods in [28, 29] and in the finite volume context in [7, 11]. More recently, this approach was extended to discontinuous Galerkin methods in [32], leading to the DG-ROD method, which restores optimal convergence rates on polygonal approximations of domains with curved boundaries. In this work, we provide a rigorous theoretical analysis of the DG-ROD method, establishing existence and uniqueness of the discrete solution and deriving error estimates for a two-dimensional linear advection-diffusion-reaction problem with homogeneous Dirichlet boundary conditions on both convex and non-convex domains. Following and extending techniques from classical finite element methods [29], we prove that, under suitable regularity assumptions on the exact solution, the DG-ROD method achieves optimal convergence despite polygonal approximations. Finally, we illustrate and confirm the theoretical results with a numerical benchmark. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_10474 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal error estimates for a discontinuous Galerkin method on curved boundaries with polygonal meshes Araújo, Adérito Santos, Milene Numerical Analysis 65N12, 65N15, 65N30 We consider a discontinuous Galerkin method for the numerical solution of boundary value problems in two-dimensional domains with curved boundaries. A key challenge in this setting is the potential loss of convergence order due to approximating the physical domain by a polygonal mesh. Unless boundary conditions can be accurately transferred from the true boundary to the computational one, such geometric approximation errors generally lead to suboptimal convergence. To overcome this limitation, a higher-order strategy based on polynomial reconstruction of boundary data was introduced for classical finite element methods in [28, 29] and in the finite volume context in [7, 11]. More recently, this approach was extended to discontinuous Galerkin methods in [32], leading to the DG-ROD method, which restores optimal convergence rates on polygonal approximations of domains with curved boundaries. In this work, we provide a rigorous theoretical analysis of the DG-ROD method, establishing existence and uniqueness of the discrete solution and deriving error estimates for a two-dimensional linear advection-diffusion-reaction problem with homogeneous Dirichlet boundary conditions on both convex and non-convex domains. Following and extending techniques from classical finite element methods [29], we prove that, under suitable regularity assumptions on the exact solution, the DG-ROD method achieves optimal convergence despite polygonal approximations. Finally, we illustrate and confirm the theoretical results with a numerical benchmark. |
| title | Optimal error estimates for a discontinuous Galerkin method on curved boundaries with polygonal meshes |
| topic | Numerical Analysis 65N12, 65N15, 65N30 |
| url | https://arxiv.org/abs/2601.10474 |