单调迭代结合虚拟区域法求解非线性障碍问题

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

讨论了二阶半线性椭圆方程障碍问题的数值求解问题.用单调迭代算法求解障碍问题,并用改进的虚拟区域法求解相关的不规则区域上具有Dirichlet边界条件的椭圆方程.在计算过程中,传统的有限元离散会导致用扩展区域规则网格计算不规则物体边界上积分的困难.为了克服此困难,给出了一种新的基于有限差分的算法,从而使得偏微分快速算法可用.算法结构简单,易于编程实现.对有扩散和增长障碍的logistic人口模型数值模拟说明算法可行且高效.

The numerical solution of obstacle problems with 2 nd-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.