A Parallel Iterative Method For Solving Elliptic Difference Equations

In this paper, we give a parallel iterative method for solving elliptic equations. We prove the convergence estimate of one dimensional and two dimensional strong diagonal dominant elliptic difference equations. In physics,the solution of non-stagnation point problem u(x,y,t) increased with t to the solution of stagnation point problem u(x, y).With the consideration, we set up the iterative parallelism relation between non-stagnation point problem and stagnation point problem. In this paper, we use the alternating explicit-implicit methods for parabolic equation on the elliptic equation, so we can get a parallel iterative method for solving elliptic equations.Consider one-dimension elliptic equation, and its difference form(e.g.N=10)the matrix form l/(h²)Cu = f. The corresponding parabolic equation:and its initial value is arbitrary. Bounded value: u(0, t) = u(l,t) = 0,0 ≤ t ≤ T. The alternating explicit-implicit methodsProm the above, we can get a iterative method for elliptic equation. Notice that G₁ and G₂ are all block diagonal matrixes, so we can compute it from floor k to k + 1 and from k + 1 to k + 2. the growing matrix from k to k + 2 isT = (I + rG₂)^-1 (I-rG₁)(I + rG₁)^-1(I- rG₂). On the iterative convergence, we can have the theorem:定理4.2. For r is small enough, we haveρ(T') < 1.So we have the iterative convergence.Consider the elliptic equation: bounded value isIts center difference form iswith the initial value, we get the following form(e.g. N=10):Its corresponding matrix form h²C_qu = f.Consider corresponding parabolic function:its initial value is arbitrary. Bounded value:u(0,t) = u(l,t) = 0. time step length is (?) = T/M, space step length is h =l/M . we get the cut space e.g. N=10,we disperse it by the alternating explicit-implicit form. We havethe growing matrix from k to k + 2 TWe give the similarity transformation of T, defineso the iterative is convergence. Otherwise, we can use the parallel computation from floor k to k+1 and from floor k + 1 to k + 2. the convergence velocity isthe convergence velocity has the same order as the SOR method.Analogous to the one-dimension elliptic equation, the two-dimension elliptic equation has similar results, with weak diagonal dominant condition, it is convergence. With strong diagonal dominant condition,the convergence velocity is...