A Finite Element Domain Decomposition Algorithm For Parabolic Equations

Parabolic equation is an important kind of partial differential equations which can de-scribe many important physical phenomena. Difference and finite element methods are two main methods to get numerical solutions for parabolic equations. Finally both of the two methods convert to solve systems of equations. Usually the dimensions of the systems are hige, but high accuracy is still required. Facing hige dimensions and high accuracy, just single computer is far away from dealing with the contradiction. Thanks to the parallel com-puters, the contradiction is alleviated. As an efficient form of computing, parallel computing is growing rapidly. Researchers have got many valuable conclusions[1,2].In the field of numerical solutions of partial differential equations, domain decomposi-tion method (DDM) is fit to apply parallel computing. The domains on which the equations are defined are always large area with high dimensions. By dividing large domain into sev-eral small subdomains, the solutions of the original problems can be translated into solving the questions on the subdomains respectively. That means large-scale computing is con-verted to small-scale computing which are independent. Because of the independence we can solve the subdomain-problems in parallel. The difficulty of the DDM lies how to deter-mine boundary values of the subdomains and the rationality of the numerical solutions. Be-sides of the above advantage of parallel computing, combining with finite element method DDM has high flexibility:firstly, local quasi-uniform grid is permitted,and the whole quasi-uniform grid isn't in need; secondly, different discrete methods can be used in different subdomains.Most DDM is about elliptic equations and is iterative. If we use iterative methods on each time levels of parabolic equations, the computing scale will be huge. According to the properties of parabolic equations, we design algorithms without iteration which are meaningful in application and theory. Dawson, etc. have done much research on DDM for parabolic equations [8,11]. They use DDM without overlapping subdomains combined with difference or finite element method to solve one and two dimensions parabolic equations. In two dimensions, they made two difference meshes:coarse and fine, then divided the domain into two subdomains.Pay attention the coarse grids are just in x direction. Using the coarse grids they formed an classical explicit procedure to solve numerical solutions on interface points. After that, the boundary values of each subdomains are known. Up to now boundary values of each subdomain are known. Finally in each subdomain they use difference or finite element method to solve the numerical solution of interior points, in parallel. Because of the coarse grids, the stable restriction is relaxed toΔt/H2≤1/6. The order of error for the algorithm is:O(Δt+h2+Hh2|lnh|+H3)Thomee have made comprehensive exposition about finite element method for parabolic equation in [6]. But the algorithm is serial, so doesn't fit to complicate domains.This paper combine finite element method and DDM to solve parabolic equation The algorithm made some modify based on [8]. We design a algorithm with non-serial structure. As an example, we increased two subdomains to four subdomains like "田". The difficulty of the algorithm is how to design a procedure to solve the numerical solution of the center point. In this paper, we introduce coarse grids in both of x and y directions to overcome the difficulty. Details as follow:First step, decompose the domain with coarse-fine grids and construct finite element space on the two mesh.Second step, solve numerical solution on the center point. Using the coarse grid in x and y directions, we can get numerical solution on the center point by an explicit proce-dure whose initial iterative value U0 is given by initial condition u0(x,y) of the parabolic equation.Third step, solve numerical solutions on interface points. Using the center and bound-ary value, we can construct "implicit in x, explicit in y " or "implicit in y, explicit in x " procedures to get the numerical value on the four interface edges.Fourth step, solve numerical solutions in subdomains. As a result of second and third steps, boundary values for each subdomain are known combining with the initial condition. Then we can construct four procedures to solve numerical solutions of interior points in four subdomains, in parallel.In the last part of the paper, we analyse the error and get the L2 norm estimate.