Minimum-residual a posteriori error estimates for hybridizable discontinuous Galerkin discretizations of the Helmholtz equation
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| Format: | Preprint |
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2023
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| _version_ | 1866910805052620800 |
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| author | Camargo, Liliana Rojas, Sergio Vega, Patrick |
| author_facet | Camargo, Liliana Rojas, Sergio Vega, Patrick |
| contents | We propose and analyze two a posteriori error indicators for hybridizable discontinuous Galerkin (HDG) discretizations of the Helmholtz equation. These indicators are built to minimize the residual associated with a local superconvergent postprocessing scheme for the primal variable, measured in a dual norm of an enlarged discrete test space. The residual minimization is reformulated into equivalent local saddle-point problems, each yielding a superconvergent postprocessed approximation of the primal variable in the asymptotic regime for sufficiently regular exact solutions and a built-in residual representation with minimal computational effort. Both error indicators are based on frequency-dependent postprocessing schemes and verify reliability and efficiency estimates for a frequency-weighted $H^1$-error for the scalar variable and the $L^2$-error for the flux. We illustrate our theoretical findings through ad-hoc numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_00418 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimum-residual a posteriori error estimates for hybridizable discontinuous Galerkin discretizations of the Helmholtz equation Camargo, Liliana Rojas, Sergio Vega, Patrick Numerical Analysis 65N12, 65N15, 65N22, 65N30, 65N50 We propose and analyze two a posteriori error indicators for hybridizable discontinuous Galerkin (HDG) discretizations of the Helmholtz equation. These indicators are built to minimize the residual associated with a local superconvergent postprocessing scheme for the primal variable, measured in a dual norm of an enlarged discrete test space. The residual minimization is reformulated into equivalent local saddle-point problems, each yielding a superconvergent postprocessed approximation of the primal variable in the asymptotic regime for sufficiently regular exact solutions and a built-in residual representation with minimal computational effort. Both error indicators are based on frequency-dependent postprocessing schemes and verify reliability and efficiency estimates for a frequency-weighted $H^1$-error for the scalar variable and the $L^2$-error for the flux. We illustrate our theoretical findings through ad-hoc numerical experiments. |
| title | Minimum-residual a posteriori error estimates for hybridizable discontinuous Galerkin discretizations of the Helmholtz equation |
| topic | Numerical Analysis 65N12, 65N15, 65N22, 65N30, 65N50 |
| url | https://arxiv.org/abs/2304.00418 |