Abstract
In this paper, we study the hp-type error estimate of the direct discontinuous Galerkin (DDG) method for the second-order elliptic problem. The discrete variational form of the DDG method for the original problem and its well-posedness are present. By introducing suitable parameters \(\beta _0, \beta _1\) in the corresponding variational form, a priori error estimates with respect to the mesh size h and the polynomial degree p in energy norm and \(L^2\) norm are derived. The convergent rates of the error with respect to the mesh size h in the energy norm and \(L^2\) norm are optimal theoretically and numerically. The suboptimal rates of the error with respect to the polynomial degree p in energy norm by half an order of p is proven theoretically. Numerical examples with singular solutions show the convergence rate about p is much greater than that obtained theoretically and it is order-doubling phenomenon when the singular point lies in the interior of the element.
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We are grateful to the editor and referees for their valuable comments and suggestions, which greatly improved the quality of our paper.
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Huang’s work is supported by the National Natural Science Foundation of China under Grant No. 11771398.
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Shang, Y., Huang, H. hp-Version direct discontinuous Galerkin method for elliptic problems. J. Appl. Math. Comput. 69, 4739–4758 (2023). https://doi.org/10.1007/s12190-023-01950-z
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DOI: https://doi.org/10.1007/s12190-023-01950-z