Volume 11, Issue 3
Two-grid hp-version Discontinuous Galerkin Finite Element Methods for Quasi-Newtonian Fluid Flows

S. Congreve & P. Houston

DOI:

Int. J. Numer. Anal. Mod., 11 (2014), pp. 496-524

Published online: 2014-11

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  • Abstract

In this article we consider the a priori and a posteriori error analysis of two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-grid method is to first solve the underlying nonlinear problem on a coarse finite element space; a fine grid solution is then computed based on undertaking a suitable linearization of the discrete problem. Here, we study two alternative linearization techniques: the first approach involves evaluating the nonlinear viscosity coefficient using the coarse grid solution, while the second method utilizes an incomplete Newton iteration technique. Energy norm error bounds are deduced for both approaches. Moreover, we design an hp-adaptive refinement strategy in order to automatically design the underlying coarse and fine finite element spaces. Numerical experiments are presented which demonstrate the practical performance of both two-grid discontinuous Galerkin methods.

  • Keywords

hp-finite element methods discontinuous Galerkin methods a posteriori error estimation adaptivity two-grid methods non-Newtonian fluids

  • AMS Subject Headings

65N30 65N55 65M60

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-496, author = {S. Congreve and P. Houston}, title = {Two-grid hp-version Discontinuous Galerkin Finite Element Methods for Quasi-Newtonian Fluid Flows}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {3}, pages = {496--524}, abstract = {In this article we consider the a priori and a posteriori error analysis of two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-grid method is to first solve the underlying nonlinear problem on a coarse finite element space; a fine grid solution is then computed based on undertaking a suitable linearization of the discrete problem. Here, we study two alternative linearization techniques: the first approach involves evaluating the nonlinear viscosity coefficient using the coarse grid solution, while the second method utilizes an incomplete Newton iteration technique. Energy norm error bounds are deduced for both approaches. Moreover, we design an hp-adaptive refinement strategy in order to automatically design the underlying coarse and fine finite element spaces. Numerical experiments are presented which demonstrate the practical performance of both two-grid discontinuous Galerkin methods.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/539.html} }
TY - JOUR T1 - Two-grid hp-version Discontinuous Galerkin Finite Element Methods for Quasi-Newtonian Fluid Flows AU - S. Congreve & P. Houston JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 496 EP - 524 PY - 2014 DA - 2014/11 SN - 11 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/ijnam/539.html KW - hp-finite element methods KW - discontinuous Galerkin methods KW - a posteriori error estimation KW - adaptivity KW - two-grid methods KW - non-Newtonian fluids AB - In this article we consider the a priori and a posteriori error analysis of two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-grid method is to first solve the underlying nonlinear problem on a coarse finite element space; a fine grid solution is then computed based on undertaking a suitable linearization of the discrete problem. Here, we study two alternative linearization techniques: the first approach involves evaluating the nonlinear viscosity coefficient using the coarse grid solution, while the second method utilizes an incomplete Newton iteration technique. Energy norm error bounds are deduced for both approaches. Moreover, we design an hp-adaptive refinement strategy in order to automatically design the underlying coarse and fine finite element spaces. Numerical experiments are presented which demonstrate the practical performance of both two-grid discontinuous Galerkin methods.
S. Congreve & P. Houston. (1970). Two-grid hp-version Discontinuous Galerkin Finite Element Methods for Quasi-Newtonian Fluid Flows. International Journal of Numerical Analysis and Modeling. 11 (3). 496-524. doi:
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