Monotone Iterations Combined With Fictitious Domain Methods for Numerical Solution of Nonlinear Obstacle Problems

RAO Ling

doi:10.21656/1000-0887.380109

Volume 39 Issue 4

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RAO Ling. Monotone Iterations Combined With Fictitious Domain Methods for Numerical Solution of Nonlinear Obstacle Problems[J]. Applied Mathematics and Mechanics, 2018, 39(4): 485-492. doi: 10.21656/1000-0887.380109

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Monotone Iterations Combined With Fictitious Domain Methods for Numerical Solution of Nonlinear Obstacle Problems

doi: 10.21656/1000-0887.380109

RAO Ling

Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R.China

Received Date: 2017-04-24
Rev Recd Date: 2017-08-09
Publish Date: 2018-04-15

Abstract

Abstract

The numerical solution of obstacle problems with 2ndorder semilinear elliptic partial differential equations (PDEs) was addressed. The nonlinear obstacle problem was solved with the monotone iteration method, and the adjoint elliptic differential equations with the Dirichlet boundary conditions on irregular domains were solved with the fictitious domain method. In the calculation process, the conventional finite element discretization resulted in the trouble of computing integrals on the irregular body boundaries with the regular mesh of the extended domain. To overcome this difficulty, a new algorithm was designed based on the finite difference method allowing the use of fast solvers for PDEs. The proposed algorithm has a simple structure and is easily programmable. The numerical simulation of a steady state problem of the logistic population model with diffusion and obstacle to growth shows that the proposed method is feasible and efficient.
- fictitious domain method,
- nonlinear obstacle problem,
- variational inequality

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References(17)

References

[1]	张石生. 变分不等式及其相关问题[M]. 重庆: 重庆出版社, 2008.(ZHANG Shisheng. Variational Inequalities and the Relevant Problem s[M]. Chongqing: Chongqing Press, 2008.(in Chinese))
[2]	DUVAUT G, LIONS J L. Inequalities in Mechanics and Physics [M]. Berlin， Heidelberg: Springer-Verlag, 1976.
[3]	KINDERLENHRER D, STANPACCHIA G. An Introduction to Variational Inequalities and Their Applications [M]. New York: Academic Press, 1980.
[4]	GLOWINSKI R. Numerical Methods for Nonlinear Variational Problems [M]. New York: Springer Verlag, 1984.
[5]	CHAN Hsinfang, FAN Chiaming, KUO Chiasen. Generalized finite difference method for solving two-dimensional non-linear obstacle problems[J]. Engineering Analysis With Boundary Elements,2013,37(9): 1189-1196.
[6]	GLOWINSKI R, KUZNETSOV Y A, PAN T W. A penalty/Newton/conjugate gradient method for the solution of obstacle problems[J]. Comptes Rendus Mathematique,2003,336(5): 435-440.
[7]	郑铁生, 李立, 许庆余. 一类椭圆型变分不等式离散问题的迭代算法[J]. 应用数学和力学, 1995,16(4): 329-335.(ZHENG Tiesheng, LI Li, XU Qingyu. An iterative method for the discrete problems of a class of elliptical variational inequalities[J]. Applied Mathematics and Mechanics,1995,16(4): 329-335.(in Chinese))
[8]	刘丹阳, 蒋娅. 混合向量变分不等式标量化及间隙函数误差界[J]. 应用数学和力学, 2017,38(6): 715-726.(LIU Danyang, JIANG Ya. Scalarization of mixed vector variational inequalities and error bounds of gap functions[J]. Applied Mathematics and Mechanics,2017, 3〖STHZ〗8(6): 715-726.(in Chinese))
[9]	郑宏, 刘德富, 李焯芬, 等. 一个新的有自由面渗流问题的变分不等式提法[J]. 应用数学和力学, 2005,26(3): 363-371.(ZHENG Hong, LIU Defu, LEE C F, et al. New variational inequality formulation for seepage problems with free surfaces[J]. Applied Mathematics and Mechanics,2005,26(3): 363-371.(in Chinese))
[10]	KORMAN P, LEUNG A W, STOJANOVIC S. Monotone iterations for nonlinear obstacle problem[J]. Journal of the Australian Mathematical Society,1990,31(3): 259-276.
[11]	PAO C V. Accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems[J]. Computers & Mathematics With Applications,2003,46(10/11): 1535-1544.
[12]	HE J W. Fictitious domain methods in fluid mechanics applications to unsteady potential flows over moving bodies[D]. PhD Thesis. Paris: University Paris 6,1994.
[13]	GLOWINSKI R, PAN T W, PERIAUX J. A one shot domain decomposition/fictitious domain method for the solution of elliptic equations[M]// Parallel Computational Fluid Dynamics: New Trends and Advances . Elsevier Science, 1993.
[14]	GILBARG D, TRUDINGER N S. Elliptic Partial Differential Equations of Second Order [M]. New York: Springer Verlag, 1977.
[15]	GLOWINSKI R, LIONS J L, TRMOLIRES R. Numerical Analysis of Variational Inequalities [M]. New York: North-Holland Publishing Company, 1976.
[16]	CHEN Hongquan, RAO Ling. The technique of the immersed boundary method: applications to the numerical solutions of imcompressible flows and wave scattering [J]. Modern Physics Letters B,2009,23(3): 437-440.
[17]	ENRIQUEZ-REMIGIO S A, ROMA A M. Incompressible flows in elastic domains: an immersed boundary method approach[J]. Applied Mathematical Modeling,2005,29(1): 35-54.