ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 04 Dec 2021 02:43:09 GMT2021-12-04T02:43:09Z5071- A generalized BPX multigrid framework covering nonnested V-cycle methodshttps://scholarbank.nus.edu.sg/handle/10635/102659Title: A generalized BPX multigrid framework covering nonnested V-cycle methods
Authors: Duan, H.-Y.; Gao, S.-Q.; Tan, R.C.E.; Zhang, S.
Abstract: More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or non-inherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far. This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper. © 2006 American Mathematical Society.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026592007-01-01T00:00:00Z
- Mixed finite elements of least-squares type for elasticityhttps://scholarbank.nus.edu.sg/handle/10635/107335Title: Mixed finite elements of least-squares type for elasticity
Authors: Duan, H.-Y.; Lin, Q.
Abstract: In terms of stress and displacement, the linear elasticity problem is discretized by a least-squares finite element method. In the case of a convex polygonal domain, the stress is approximated by the lowest-order Raviart-Thomas-Nédélec flux element, and the displacement by the linear C0 element. We obtain coerciveness and optimal H1, L2 and H(div)-error bounds, uniform in Lamé constant λ, for displacement and stress, respectively. Our method also allows the use of any other combination of conforming elements for stress and displacement, e.g., C0 elements for all variables. © 2004 Elsevier B.V. All rights reserved.
Fri, 18 Mar 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1073352005-03-18T00:00:00Z
- The local L2 projected C0 finite el ement method for Maxwell problemhttps://scholarbank.nus.edu.sg/handle/10635/104312Title: The local L2 projected C0 finite el ement method for Maxwell problem
Authors: Duan, H.-Y.; Jia, F.; Lin, P.; Tan, R.C.E.
Abstract: An element-local L2-projected C0 finite element method is presented to approximate the nonsmooth solution being not in H1 of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local L2 projectors are applied to both curl and div operators. The C0 linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in L2 norm is established uniform in the mesh-size, and the condition number O(h-2) of the resulting linear system is proven. For the solution and its curl in Hr with r < 1 we obtain an error bound O(hr) in an energy norm. Numerical experiments confirm the theoretical error bound. © 2009 Societ y for Industrial and Applied Mathematics.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043122009-01-01T00:00:00Z
- A least-squares finite element method for the magnetostatic problem in a multiply connected Lipschitz domainhttps://scholarbank.nus.edu.sg/handle/10635/102668Title: A least-squares finite element method for the magnetostatic problem in a multiply connected Lipschitz domain
Authors: Duan, H.-Y.; Lin, P.; Saikrishnan, P.; Tan, R.C.E.
Abstract: A new least-squares finite element method is developed for the curl-div magnetostatic problem in Lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials. In order to deal with possibly low regularity of the solution, local L2 projectors are introduced to standard least-squares formulation and applied to both curl and div operators. Coercivity is established by adding suitable mesh-dependent bilinear terms. As a result, any continuous finite elements (lower-order elements are enriched with suitable bubbles) can be employed. A desirable error bound is obtained: ∥u - uh∥0 ≤ C ∥u - ũ∥0, where uh and ũ are the finite element approximation and the finite element interpolant of the exact solution u, respectively. Numerical tests confirm the theoretical results. © 2007 Society for Industrial and Applied Mathematics.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026682007-01-01T00:00:00Z
- Erratum: A generalized BPX multigrid framework covering nonnested V-cycle methods (Mathematics of Computation (2007) 76:257 (137-152))https://scholarbank.nus.edu.sg/handle/10635/104681Title: Erratum: A generalized BPX multigrid framework covering nonnested V-cycle methods (Mathematics of Computation (2007) 76:257 (137-152))
Authors: Duan, H.-Y.; Gao, S.-Q.; Tan, R.C.E.; Zhang, S.
Mon, 01 Oct 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1046812007-10-01T00:00:00Z
- Analysis of a least-squares finite element method for the thin plate problemhttps://scholarbank.nus.edu.sg/handle/10635/102855Title: Analysis of a least-squares finite element method for the thin plate problem
Authors: Duan, H.-y.; Gao, S.-q.; Jiang, B.-n.; Tan, R.C.E.
Abstract: A new least-squares finite element method is analyzed for the thin plate problem subject to various boundary conditions (clamped, simply supported and free). The unknown variables are deflection, slope, moment and shear force. The coercivity property is established. As a result, all variables can be approximated by any conforming finite elements. In particular, an H1-ellipticity is proven for the free thin plate. This indicates that optimal error bounds hold for all variables with the use of equal-order continuous elements. Numerical experiments are performed to confirm the theoretical results obtained. © 2008 IMACS.
Fri, 01 May 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028552009-05-01T00:00:00Z
- L2-projected least-squares finite element methods for the Stokes equationshttps://scholarbank.nus.edu.sg/handle/10635/103472Title: L2-projected least-squares finite element methods for the Stokes equations
Authors: Duan, H.-Y.; Lin, P.; Saikrishnan, P.; Tan, R.C.E.
Abstract: Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition, A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results. © 2006 Society for Industrial and Applied Mathematics.
Sun, 01 Jan 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034722006-01-01T00:00:00Z