{
  "annotation_id": "835e60e2-e50e-4322-81e2-ee6d17535bae",
  "date_created": "2026-02-17T05:53:24.340000Z",
  "date_modified": "2026-02-17T05:53:24.340000Z",
  "file_hash": "fe3c3cdc62076e624c7c64c795921cf375cad2524515d0ab40639d46d006fba7",
  "private": false,
  "record": {
    "abstract": "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.",
    "arxiv_id": "2601.10474",
    "authors": [
      "Ad\u00e9rito Ara\u00fajo",
      "Milene Santos"
    ],
    "categories": [
      "math.NA",
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Optimal error estimates for a discontinuous Galerkin method on curved boundaries with polygonal meshes",
    "url": "https://arxiv.org/abs/2601.10474",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "7d4fc49a-5110-43b5-b8ca-eee82c223ac5",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}