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Integral-equation methods for inhomo...
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Askham, Travis.
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Integral-equation methods for inhomogeneous elliptic partial differential equations in complex geometry.
紀錄類型:
書目-電子資源 : Monograph/item
正題名/作者:
Integral-equation methods for inhomogeneous elliptic partial differential equations in complex geometry./
作者:
Askham, Travis.
面頁冊數:
129 p.
附註:
Source: Dissertation Abstracts International, Volume: 77-07(E), Section: B.
Contained By:
Dissertation Abstracts International77-07B(E).
標題:
Mathematics. -
電子資源:
http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10025674
ISBN:
9781339521275
Integral-equation methods for inhomogeneous elliptic partial differential equations in complex geometry.
Askham, Travis.
Integral-equation methods for inhomogeneous elliptic partial differential equations in complex geometry.
- 129 p.
Source: Dissertation Abstracts International, Volume: 77-07(E), Section: B.
Thesis (Ph.D.)--New York University, 2016.
Over the last several decades, integral-equation methods have emerged as powerful tools for the solution of the homogeneous partial differential equations of mathematical physics, including the Laplace, Helmholtz and Maxwell equations. By using the methods of potential theory, boundary value problems can be recast as boundary integral equations, reducing the dimensionality of the problem by one and permitting geometrically flexible discretization. Combined with suitable fast algorithms, this has led to optimal or nearly optimal complexity solvers in a variety of important application areas. For inhomogeneous, variable coefficient and nonlinear equations, there is no corresponding reduction in dimensionality, since the interior of the domain needs to be discretized and integral-equation methods appear to be less natural. They lead to dense linear systems of equations, while direct discretization of the partial differential equation leads to a sparse system of the same size.
ISBN: 9781339521275Subjects--Topical Terms:
515831
Mathematics.
Integral-equation methods for inhomogeneous elliptic partial differential equations in complex geometry.
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Source: Dissertation Abstracts International, Volume: 77-07(E), Section: B.
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Adviser: Leslie Greengard.
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Thesis (Ph.D.)--New York University, 2016.
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Over the last several decades, integral-equation methods have emerged as powerful tools for the solution of the homogeneous partial differential equations of mathematical physics, including the Laplace, Helmholtz and Maxwell equations. By using the methods of potential theory, boundary value problems can be recast as boundary integral equations, reducing the dimensionality of the problem by one and permitting geometrically flexible discretization. Combined with suitable fast algorithms, this has led to optimal or nearly optimal complexity solvers in a variety of important application areas. For inhomogeneous, variable coefficient and nonlinear equations, there is no corresponding reduction in dimensionality, since the interior of the domain needs to be discretized and integral-equation methods appear to be less natural. They lead to dense linear systems of equations, while direct discretization of the partial differential equation leads to a sparse system of the same size.
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In this thesis, we present a new collection of integral-equation methods for inhomogeneous partial differential equations including the Poisson, modified Helmholtz, and modified Stokes equations that are of optimal computational complexity, while fully adaptive, high order accurate and easy to use even in complex geometry. For this, we have combined fast methods for computing the volume integral operators of potential theory, with new quadrature methods for layer potentials, and a new method for smooth function extension from arbitrarily-shaped domains. Unlike finite difference and finite element discretization of the governing partial differential equation, the only unknowns in the formulation lie on the domain boundary, so that there is still an effective reduction in dimensionality in terms of the size of the linear system which needs to be solved. The interior degrees of freedom are accounted for using an explicit integral transform, for which we have developed specialized versions of the fast multipole method. These tools can be extended in a straightforward manner to create fast, high order accurate solvers for more complex problems, including variable coefficient problems and the incompressible Navier-Stokes equations.
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