Cecilia.soriano: Cecilia.soriano moved page Draft Soriano 615706486 to Drzisga et al 2019a

2020-02-21T09:07:21Z

Cecilia.soriano moved page Draft Soriano 615706486 to Drzisga et al 2019a

Cecilia.soriano at 09:06, 21 February 2020

2020-02-21T09:06:39Z

Cecilia.soriano at 09:03, 21 February 2020

2020-02-21T09:03:23Z

Cecilia.soriano: Created page with "==1 Title, abstract and keywords

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	We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [''Appl. Numer. Math.'', 122 (2017), pp. 14--38]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piecewise smooth approximation of the matrices involved in a standard finite element discretization. In particular, it relies on the projection of smooth so-called stencil functions onto high-order polynomial subspaces. The performance advantage of the surrogate matrix methodology is seen in constructions where each stencil function uniquely determines the values of a significant collection of matrix entries. Such constructions are shown to be widely achievable through the use of locally structured meshes. Therefore, this methodology can be applied to a wide variety of physically meaningful problems, including nonlinear problems and problems with curvilinear geometries. Rigorous a priori error analysis certifies the convergence of a novel surrogate method for the variable coefficient Poisson equation. The flexibility of the methodology is also demonstrated through the construction of novel methods for linear elasticity and nonlinear diffusion problems. In numerous numerical experiments, we demonstrate the efficacy of these new methods in a matrix-free environment with geometric multigrid solvers. In our experiments, up to a twenty-fold decrease in computation time is witnessed over the classical method with an otherwise identical implementation		We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [''Appl. Numer. Math.'', 122 (2017), pp. 14--38]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piecewise smooth approximation of the matrices involved in a standard finite element discretization. In particular, it relies on the projection of smooth so-called stencil functions onto high-order polynomial subspaces. The performance advantage of the surrogate matrix methodology is seen in constructions where each stencil function uniquely determines the values of a significant collection of matrix entries. Such constructions are shown to be widely achievable through the use of locally structured meshes. Therefore, this methodology can be applied to a wide variety of physically meaningful problems, including nonlinear problems and problems with curvilinear geometries. Rigorous a priori error analysis certifies the convergence of a novel surrogate method for the variable coefficient Poisson equation. The flexibility of the methodology is also demonstrated through the construction of novel methods for linear elasticity and nonlinear diffusion problems. In numerous numerical experiments, we demonstrate the efficacy of these new methods in a matrix-free environment with geometric multigrid solvers. In our experiments, up to a twenty-fold decrease in computation time is witnessed over the classical method with an otherwise identical implementation
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-==1 Title, abstract and keywords<!-- Your document should start with a concise and informative title. Titles are often used in information-retrieval systems. Avoid abbreviations and formulae where possible. Capitalize the first word of the title.
+==Abstract <!-- Your document should start with a concise and informative title. Titles are often used in information-retrieval systems. Avoid abbreviations and formulae where possible. Capitalize the first word of the title.  Provide a maximum of 6 keywords, and avoiding general and plural terms and multiple concepts (avoid, for example, 'and', 'of'). Be sparing with abbreviations: only abbreviations firmly established in the field should be used. These keywords will be used for indexing purposes.  An abstract is required for every document; it should succinctly summarize the reason for the work, the main findings, and the conclusions of the study. Abstract is often presented separately from the article, so it must be able to stand alone. For this reason, references and hyperlinks should be avoided. If references are essential, then cite the author(s) and year(s). Also, non-standard or uncommon abbreviations should be avoided, but if essential they must be defined at their first mention in the abstract itself. -->==
-Provide a maximum of 6 keywords, and avoiding general and plural terms and multiple concepts (avoid, for example, 'and', 'of'). Be sparing with abbreviations: only abbreviations firmly established in the field should be used. These keywords will be used for indexing purposes.
+We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [''Appl. Numer. Math.'', 122 (2017), pp. 14--38]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piecewise smooth approximation of the matrices involved in a standard finite element discretization. In particular, it relies on the projection of smooth so-called stencil functions onto high-order polynomial subspaces. The performance advantage of the surrogate matrix methodology is seen in constructions where each stencil function uniquely determines the values of a significant collection of matrix entries. Such constructions are shown to be widely achievable through the use of locally structured meshes. Therefore, this methodology can be applied to a wide variety of physically meaningful problems, including nonlinear problems and problems with curvilinear geometries. Rigorous a priori error analysis certifies the convergence of a novel surrogate method for the variable coefficient Poisson equation. The flexibility of the methodology is also demonstrated through the construction of novel methods for linear elasticity and nonlinear diffusion problems. In numerous numerical experiments, we demonstrate the efficacy of these new methods in a matrix-free environment with geometric multigrid solvers. In our experiments, up to a twenty-fold decrease in computation time is witnessed over the classical method with an otherwise identical implementation
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← Older revision	Revision as of 09:07, 21 February 2020
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Drzisga et al 2019a - Revision history

Cecilia.soriano: Cecilia.soriano moved page Draft Soriano 615706486 to Drzisga et al 2019a

Cecilia.soriano at 09:06, 21 February 2020

Cecilia.soriano at 09:03, 21 February 2020

Cecilia.soriano: Created page with "==1 Title, abstract and keywords