Analysis of a quasicontinuum method in one dimension
The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic mode...
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| Format: | Journal article |
| Language: | English |
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2008
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| _version_ | 1826274591375360000 |
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| author | Ortner, C Suli, E |
| author_facet | Ortner, C Suli, E |
| author_sort | Ortner, C |
| collection | OXFORD |
| description | The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard-Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete -norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a 'nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations. © EDP Sciences. |
| first_indexed | 2024-03-06T22:45:47Z |
| format | Journal article |
| id | oxford-uuid:5d205d4e-1f24-4a87-bc8e-496ee184effc |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:45:47Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:5d205d4e-1f24-4a87-bc8e-496ee184effc2022-03-26T17:32:25ZAnalysis of a quasicontinuum method in one dimensionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5d205d4e-1f24-4a87-bc8e-496ee184effcEnglishSymplectic Elements at Oxford2008Ortner, CSuli, EThe quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard-Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete -norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a 'nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations. © EDP Sciences. |
| spellingShingle | Ortner, C Suli, E Analysis of a quasicontinuum method in one dimension |
| title | Analysis of a quasicontinuum method in one dimension |
| title_full | Analysis of a quasicontinuum method in one dimension |
| title_fullStr | Analysis of a quasicontinuum method in one dimension |
| title_full_unstemmed | Analysis of a quasicontinuum method in one dimension |
| title_short | Analysis of a quasicontinuum method in one dimension |
| title_sort | analysis of a quasicontinuum method in one dimension |
| work_keys_str_mv | AT ortnerc analysisofaquasicontinuummethodinonedimension AT sulie analysisofaquasicontinuummethodinonedimension |