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Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

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Abstract

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space \(V({{\mathcal {T}_{H}}},\boldsymbol {P})\). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space \(V({{\mathcal {T}_{h}}},\boldsymbol {p})\); thereby, only a linear system of equations is solved on the richer space \(V({{\mathcal {T}_{h}}},\boldsymbol {p})\). In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces \(V({{\mathcal {T}_{H}}},\boldsymbol {P})\) and \(V({{\mathcal {T}_{h}}},\boldsymbol {p})\), respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the CPU time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method.

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Acknowledgements

PH acknowledges the financial support of the EPSRC under the grant EP/H005498. TW acknowledges the financial support of the Swiss National Science Foundation (SNF) under grant No. 200021126594.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Scott Congreve & Paul Houston

  2. Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Thomas P. Wihler

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  1. Scott Congreve
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  2. Paul Houston
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  3. Thomas P. Wihler
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Correspondence to Paul Houston.

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Congreve, S., Houston, P. & Wihler, T.P. Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs. J Sci Comput 55, 471–497 (2013). https://doi.org/10.1007/s10915-012-9644-1

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