Research On Numerical Methods To Variational Inequalities

We discuss two blocks in this dissertation. First, we propose two algorithms for solving the obstacle problem; Second, we investigate the convergence rate of relaxation method for solving a simple friction problem. We expand theory analysis on these two models in mathematical technology and engineering thinking. Some important conclusions are drawn. Finally, some numerical examples are illustrated.The obstacle problem is the typical example of the elliptic variational inequality of the first kind. For the computation of approximate solutions, we propose two algorithms in different ways. The purpose of one algorithm is to try finding the approximated region of the contact domain Ω₁ by iteration. In each iteration step, we need solving linear systems. The following useful algorithm is presented: Solving Au⁰ = f, we get the initial position of the elastic membrane u°. Then for n = 0,1,2, ? ? ? and small ε > 0,i) Computing d_i(n) = u_iⁿ-ψ(X_i), i = 1, ? ? ?, N and d_minⁿ =min(?) d_i(n)ii) If d_minⁿ 0 STOP, elseiii) For all i ∈ Ω₁ⁿ , modify the matrix A and the load vector f byiv) Solve Auⁿ⁺¹ = f.Thus we think the nodes are contact nodes if d_minⁿ ≤ d_i(n) ≤ d_minⁿ+ε. Finally, some numerical examples are reported.The other algorithm is presented for solving the corresponding complementary problem. We begin with a suitable initial guess value of the solution, then construct a iteration algorithm to generate the monotone sequence. The algorithm is following:i) Given the initial guess value u⁰ satisfies the inequalities...