• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.23-40
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• 1998

Mixed finite element method is developed to approxi-mate the solution of the initial-boundary value problem for a strongly nonlinear second-order hyperbolic equation in divergence form. Exis-tence and uniqueness of the approximation are proved and optimal-order $L\infty$-in-time $L^2$-in-space a priori error estimates are derived for both the scalar and vector functions approximated by the method.

• East Asian mathematical journal
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• v.30 no.3
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• pp.327-334
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• 2014

In this paper, we construct extrapolated expanded mixed finite element approximations to approximate the scalar unknown, its gradient and its flux of semilinear Sobolev equations. To avoid the difficulty of solving the system of nonlinear equations, we use an extrapolated technique in our construction of the approximations. Some numerical examples are used to show the efficiency of our schemes.

• Structural Engineering and Mechanics
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• v.13 no.3
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• pp.277-298
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• 2002

In this study, free vibration analysis of Reissner plates on Pasternak foundation is carried out by mixed finite element method based on the G$\hat{a}$teaux differential. New boundary conditions are established for plates on Pasternak foundation. This method is developed and applied to numerous problems by Ak$\ddot{o}$z and his co-workers. In dynamic analysis, the problem reduces to the solution of a standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and bending and torsional moments, transverse shear forces, rotations and displacements are the basic unknowns. The element performance is assessed by comparison with numerical examples known from literature. Validity limits of Kirchhoff plate theory is tested by dynamic analysis. Shear locking effects are tested as far as $h/2a=10^{-6}$ and it is observed that REC32 is free from shear locking.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.549-569
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• 2012

In this paper asymptotic error expansions for mixed finite element approximations to a class of second order elliptic optimal control problems are derived under rectangular meshes, and the Richardson extrapolation of two different schemes and interpolation defect correction can be applied to increase the accuracy of the approximations. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators of the mixed finite element method for optimal control problems.

• Journal of the Korean Mathematical Society
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• v.51 no.3
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• pp.545-565
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• 2014

In this paper we derive a priori $L^{\infty}(L^2)$ error estimates for expanded mixed finite element formulations of semilinear Sobolev equations. This formulation expands the standard mixed formulation in the sense that three variables, the scalar unknown, the gradient and the flux are explicitly treated. Based on this method we construct finite element semidiscrete approximations and fully discrete approximations of the semilinear Sobolev equations. We prove the existence of semidiscrete approximations of u, $-{\nabla}u$ and $-{\nabla}u-{\nabla}u_t$ and obtain the optimal order error estimates in the $L^{\infty}(L^2)$ norm. And also we construct the fully discrete approximations and analyze the optimal convergence of the approximations in ${\ell}^{\infty}(L^2)$ norm. Finally we also provide the computational results.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.1-28
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• 2004

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by an $H^1$-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index one. A priori error estimates for semidiscrete scheme are derived for both differ-ential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.137-146
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• 2017

The size-dependent behavior of single walled carbon nanotubes (SWCNT) embedded in the elastic medium and subjected to the initial axial force is investigated using the mixed finite element method. The SWCNT is assumed to be Euler-Bernoulli beam incorporating nonlocal theory developed by Eringen. The mixed finite element model shows its great advantage of dealing with nonlocal behavior of SWCNT subjected to a concentrated load owing to the existence of two coefficients ${\alpha}_1$ and ${\alpha}_2$. This is the first numerical approach to deal with a puzzling fact of nonlocal theory with concentrated load. Numerical examples are performed to show the accuracy and efficiency of the present method. In addition, parametric study is carefully carried out to point out the influences of nonlocal effect, the elastic medium, and the initial axial force on the behavior of the carbon nanotubes.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.61-82
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• 2000

In this paper two so-called regularized Green's functions are introduced to derive the optimal maximum norm error estimates for the unknown function and the adjoint vector-valued function for mixed finite element methods of Laplacian operator. One contribution of the paper is a demonstration of how the boundedness of $L^1$-norm estimate for the second Green's function ${\lambda}_2$ and the optimal maximum norm error estimate for the adjoint vector-valued function are proved. These results are seemed to be to be new in the literature of the mixed finite element methods.

• Korean Journal of Mathematics
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• v.13 no.2
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• pp.127-137
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• 2005

The objective of numerical analysis is to devise and analyze efficient algorithms or numerical methods for equations arising in mathematical modeling for science and engineering. In this article, we present some recent topics in computational mathematics, specially in the finite element method and overview the development of the mixed finite element method in the context of second order elliptic and parabolic problems. Multiscale methods such as MsFEM, HMM, and VMsM are included.

• Computational Structural Engineering
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• v.7 no.2
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• pp.111-120
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• 1994

Cracks in rubber bonded to a rigid material such as steel are analyzed with the aid of a mixed finite element technique. Firstly the weak form is derived for finite element analysis of an incompressible material, and the Mooney-Rivlin form is assumed for the constitutive modeling of rubber. The numerical results from finite element analysis is examined to confirm the accuracy and convergence of solution by way of comparison to other numerical results. The interpretation of the J-integral for large elastic deformation as the energy release rate is confirmed, and the J-integral is calculated for varing crack length. The crack growth stability is discussed using the result of finite element analysis.