A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems
We develop a calculus for nonlocal operators that mimics Gauss's theorem and Green's identities of the classical vector calculus. The operators we define do not involve derivatives. We then apply the nonlocal calculus to define weak formulations of nonlocal "boundary-value" probl...
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| Veröffentlicht in: | Multiscale modeling & simulation 2010-01, Vol.8 (5), p.1581-1598 |
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| container_title | Multiscale modeling & simulation |
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| creator | Gunzburger, Max Lehoucq, R B |
| description | We develop a calculus for nonlocal operators that mimics Gauss's theorem and Green's identities of the classical vector calculus. The operators we define do not involve derivatives. We then apply the nonlocal calculus to define weak formulations of nonlocal "boundary-value" problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal problems, we derive a fundamental solution and Green's functions, demonstrate that weak formulations of the nonlocal "boundary-value" problems are well posed, and show how, under appropriate limits, the nonlocal problems reduce to their local analogues. [PUBLICATION ABSTRACT] |
| doi_str_mv | 10.1137/090766607 |
| format | Article |
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| identifier | ISSN: 1540-3459 |
| ispartof | Multiscale modeling & simulation, 2010-01, Vol.8 (5), p.1581-1598 |
| issn | 1540-3459 1540-3467 |
| language | eng |
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| source | SIAM Journals Online |
| subjects | Boundary value problems Calculus Derivatives Dirichlet problem Mathematical analysis Mathematical models Normal distribution Operators Partial differential equations Studies Vectors (mathematics) |
| title | A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems |
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