Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

Gergina V. Pencheva; Martin Vohralík; Mary F. Wheeler; Tim Wildey

doi:10.1137/110839047

Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

Gergina V. Pencheva, Martin Vohralík, Mary F. Wheeler, Tim Wildey

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25 Scopus citations

Abstract

We consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h, whereas the interfaces are partitioned by a mesh of much coarser size H, and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested. © 2013 Society for Industrial and Applied Mathematics.

Original language	English (US)
Pages (from-to)	526-554
Number of pages	29
Journal	SIAM Journal on Numerical Analysis
Volume	51
Issue number	1
DOIs	https://doi.org/10.1137/110839047
State	Published - Jan 2013
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): (KAUST)-AEA-UTA08-687
Acknowledgements: A portion of this research was supported by the U.S.Department of Energy, Office of Science, Office of Basic Energy Sciences. The Center for Frontiersof Subsurface Energy Security (CFSES) is a DOE Energy Frontier Research Center, under AwardNumber DE-SC0001114. The authors gratefully acknowledge the financial support provided by theNSF-CDI under contract DMS 0835745 and King Abdullah University of Science and Technology(KAUST)-AEA-UTA08-687 and DOE grant DE-FGO2-04ER25617. This author was supported by the GNR MoMaS project “Numerical Simulations and Mathematical Modeling ofUnderground Nuclear Waste Disposal,” PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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title = "Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling",

abstract = "We consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h, whereas the interfaces are partitioned by a mesh of much coarser size H, and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested. {\textcopyright} 2013 Society for Industrial and Applied Mathematics.",

author = "Pencheva, {Gergina V.} and Martin Vohral{\'i}k and Wheeler, {Mary F.} and Tim Wildey",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): (KAUST)-AEA-UTA08-687 Acknowledgements: A portion of this research was supported by the U.S.Department of Energy, Office of Science, Office of Basic Energy Sciences. The Center for Frontiersof Subsurface Energy Security (CFSES) is a DOE Energy Frontier Research Center, under AwardNumber DE-SC0001114. The authors gratefully acknowledge the financial support provided by theNSF-CDI under contract DMS 0835745 and King Abdullah University of Science and Technology(KAUST)-AEA-UTA08-687 and DOE grant DE-FGO2-04ER25617. This author was supported by the GNR MoMaS project “Numerical Simulations and Mathematical Modeling ofUnderground Nuclear Waste Disposal,” PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - We consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h, whereas the interfaces are partitioned by a mesh of much coarser size H, and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested. © 2013 Society for Industrial and Applied Mathematics.

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Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

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