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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 35
COMPUTATIONAL METHODS FOR ENGINEERING TECHNOLOGY Edited by: B.H.V. Topping and P. Iványi
Chapter 8
On a Class of Inexact Type Methods Based on Generic Approximate Sparse Inverses Multigrid Method for Solving Nonlinear Partial Differential Equations G.A. Gravvanis and C.K. FilelisPapadopoulos
Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, Xanthi, Greece G.A. Gravvanis, C.K. FilelisPapadopoulos, "On a Class of Inexact Type Methods Based on
Generic Approximate Sparse Inverses Multigrid Method
for Solving Nonlinear Partial Differential Equations", in B.H.V. Topping and P. Iványi, (Editor), "Computational Methods for Engineering Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 8, pp 191221, 2014. doi:10.4203/csets.35.8
Keywords: nonlinear partial differential equations, inexact Newton’s method,
inexact Broyden’s method, multigrid method, generalized approximate sparse
inverses, finite difference method, inexact Jacobianfree Newtonmultigrid, full
approximation scheme, Krylov accelerated multigrid methods.
Abstract
Many engineering and scientific problems are described by nonlinear partial
differential equations. This category of equations represents a large class of
commonly occurring problems in Mathematical Sciences and Engineering. The
numerical solution of nonlinear Partial Differential Equations requires a linearization
method such as Inexact Newton’s or Inexact Broyden’s method and an efficient
solver to compute the approximate solution in each step. Multigrid methods are
considered efficient methods of near optimal complexity and in conjunction with
Approximate Sparse Inverses have been experimentally proven to be effective for
various classes of problems. Herewith, the Generic Approximate Sparse Inverse,
based on Approximate Inverse Sparsity patterns, is used in conjunction with the
multigrid method as an inner solver for Inexact Newton’s and Inexact Broyden’s
method. Moreover, two new schemes based on Inexact Newton’s and Inexact
Broyden’s method, in conjunction with the multigrid method based on Generic
Approximate Sparse Inverses, are proposed. In the proposed schemes the
approximate inverse is explicitly computed during the first iteration and is then
updated for the following iterations until convergence. The update of the
approximate inverse is carried out with a modified version of the Broyden’s formula
targeting only the nonzeros. Furthermore, the JacobianFree NewtonMultigrid in
conjunction with Generic Approximate Sparse Inverses is presented. The Jacobian
Free NewtonMultigrid method does not require the Jacobian matrix explicitly,
instead the Jacobian ‘times’ a vector is approximated by finite differences.
Moreover, the Full Approximation Scheme for nonlinear systems of equations in
conjunction with Generic Approximate Sparse Inverses is presented. The Full
Approximation Scheme applies multigrid techniques to the solution of systems of
nonlinear equations. Implementation details and discussions concerning the various
proposed schemes are provided.
In order to increase robustness of the multigrid
algorithms, the proposed schemes are accelerated with the use of a Krylov subspace
iterative method. Moreover, the proposed schemes are compared by solving the Bratu problem and numerical results concerning the performance and convergence
behaviour are given.
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