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Now showing 1 - 6 of 6

Analysis of least squares finite element methods for a parameter-dependent first-order system
(1998-01-01)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A parameter-dependent first-order system arising from elasticity problems is introduced. The system corresponds to the 2D isotropic elasticity equations under a stress-pressure-displacement formulation in which the nonnegative parameter measures the material compressibility for the elastic body. Standard and weighted least squares finite element methods are applied to this system, and analyses for different values of the parameter are performed in a unified manner. The methods offer certain advantages such as they need not satisfy the Babuska-Brezzi condition, a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, the resulting algebraic system is symmetric and positive definite, accurate approximations of all the unknowns can be obtained simultaneously, and, especially, computational results do not exhibit any significant numerical locking as the parameter tends to zero which corresponds to the incompressible elasticity problem (or equivalently, the Stokes problem). With suitable boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various values of the parameter are given to demonstrate the theoretical estimates.
Least-squares finite element methods for the elasticity problem
(1997-12-18)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.
A two-stage least-squares finite element method for the stress-pressure-displacement elasticity equations
(1998-05-01)
Yang, SY
;
Chang, CL
;
應用數學系
;
Department of Applied Mathematics
A new stress-pressure-displacement formulation for the planar elasticity equations is proposed by introducing the auxiliary variables, stresses, and pressure. The resulting first-order system involves a nonnegative parameter that measures the material compressibility for the elastic body. A two-stage least-squares finite element procedure is introduced for approximating the solution to this system with appropriate boundary conditions. It is shown that the two-stage least-squares scheme is stable and, with respect to the order of approximation for smooth exact solutions, the rates of convergence of the approximations for all the unknowns are optimal both in the H-1-norm and in the L-2-norm. Numerical experiments with various values of the parameter are examined, which demonstrate the theoretical estimates. Among other things, computational results indicate that the behavior of convergence is uniform in the nonnegative parameter. (C) 1998 John Wiley & Sons, Inc.
On the convergence and stability of the standard least squares finite element method for first-order elliptic systems
(1998-07-01)
Yang, SY
;
應用數學系
;
Department of Applied Mathematics
A general framework of the theoretical analysis for the convergence and stability of the standard least squares finite element approximations to boundary value problems of first-order linear elliptic systems is established in a natural norm. With a suitable density assumption, the standard least squares method is proved to be convergent without requiring extra smoothness of the exact solutions. The method is also shown to be stable with respect to the natural norm. Some representative problems such as the grad-div type problems and the Stokes problem are demonstrated. (C) 1998 Published by Elsevier Science Inc. All rights reserved.
Least-squares finite element methods for the elasticity problem
(1997-12-18)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.
Analysis of least squares finite element methods for a parameter-dependent first-order system
(1998-01-01)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A parameter-dependent first-order system arising from elasticity problems is introduced. The system corresponds to the 2D isotropic elasticity equations under a stress-pressure-displacement formulation in which the nonnegative parameter measures the material compressibility for the elastic body. Standard and weighted least squares finite element methods are applied to this system, and analyses for different values of the parameter are performed in a unified manner. The methods offer certain advantages such as they need not satisfy the Babuska-Brezzi condition, a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, the resulting algebraic system is symmetric and positive definite, accurate approximations of all the unknowns can be obtained simultaneously, and, especially, computational results do not exhibit any significant numerical locking as the parameter tends to zero which corresponds to the incompressible elasticity problem (or equivalently, the Stokes problem). With suitable boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various values of the parameter are given to demonstrate the theoretical estimates.

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