Numerical solution for a nonlinear obstacle problem
- Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China.
- Center for General Educatin, China Medical University, Taichung, 40402,, Taiwan.
A monotone iterations algorithm combined with the finite difference method is constructed for an obstacle
problem with semilinear elliptic partial differential equations of second order. By means of Dirac delta
function to improve the computation procedure of the
discretization, the finite difference method is still practicable even though the obstacle boundary is irregular. The numerical simulations show that our proposed methods are feasible and effective for the nonlinear obstacle problem.
- Finite difference method
- nonlinear obstacle problem
- variational inequality
- elliptic partial differential equation
D. Boffi, L. Gastaldi, A finite element approach for the immersed boundary method, Comput. Structures, 81 (2003), 491–501.
H.-F. Chan, C.-M. Fan, C.-W. Kuo, Generalized finite difference method for solving two-dimensional non-linear obstacle problems, Eng. Anal. Bound. Elem., 37 (2013), 1189–1196.
S. A. Enriquez-Remigio, A. M. Roma, Incompressible flows in elastic domains: an immersed boundary method approach, Appl. Math. Model., 29 (2005), 35–54.
R. Glowinski, J.-L. Lions, R. Tremolieres, Numerical analysis of variational inequalities , North-Holland Publishing Co., Amsterdam-New York (1976)
R. Glowinski, T.-W. Pan, J. Periaux, A fictitious domain method for Dirichlet problem and applications, Comput. Methods Appl. Mech. Engrg., 111 (1994), 283–303.
P. Korman, A. W. Leung, S. Stojanovic, Monotone iterations for nonlinear obstacle problem, J. Austral. Math. Soc. Ser. B, 31 (1990), 259–276.
C. S. Peskin, Numerical analysis of blood flow in the heart, J. Computational Phys., 25 (1977), 220–252.
C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479–517.
L. Rao, H. Q. Chen, The technique of the immersed boundary method: application to solving shape optimization problem, J. Appl. Math. Phy., 5 (2017), 329–340.
V. K. Saulev, On the solution of some boundary value problems on high performance computers by fictitious domain method, Siberian Math. J., 4 (1963), 912–925.
A. L. F. L. E. Silva, A. Silveira-Neto, J. J. R. Damasceno , Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J. Computational Phys., 189 (2003), 351–370.