The result of constructing two domain decomposition methods for solving biharmonic problem

Tạp chí Khoa học & Công nghệ - số 2(50)/năm 2009 Toán, Thống kê –KH tự nhiên –KH máy tính

1

THE RESULT OF CONSTRUCTING TWO DOMAIN DECOMPOSITION

METHODS FOR SOLVING BIHARMONIC PROBLEM

Truong Ha Hai – Vu Vinh Quang (Faculty of Information and Communication Technology – Thainguyen University),

Do Diep Anh (University of Sciences - Thainguyen University)

1. The domain decomposition method for solving the second elliptic problem

Figure 1

Let be a domain in

2 R

with boundary being Lipschitz continuousness. To consider

problem:

, .

, ,

u x

u f x

, In which

2 1/ 2 f L H ( ), ( ). (1)

To suppose that domain is divided into two non-intersection subdomains 1, 2,

signing

1 2 1 1 2 2 , \ , \ .

To suppose is Lipschitz boundary, too.

Calling ui is value of solution, u, of problem (1) in domain i, n

i

outer normal direction on

( 1, 2). i

i

The problem (1) can rewrite the following many domains:

1 1

1 1

1 2

2 1

2 1

2 2

2 2

, ,

, ,

, ,

, ,

, ,

, .

u f x

u x

u u x

u u

x

n n

u x

u f x

(2)

The 3 and 4 equations in (2) are transmission conditions on boundary, describing

continuous condition of function and derivative as it varies through common boundary . The

solution of two problems in two subdomains must satisfy condition (2). To define condition on

decomposition boundary play an important role in solving the problems in subdomains. In last

years, it is two following research directions:

Saito – Fujita method (2001)

From idea about specified function value on decomposition boundary, Norikazu Saito –

Hiroshi Fujita had proposed iterative scheme for determining function value as following:

Notation

( ) ( )

1 2 ,

k k

u u

is convergent function chains to

1 2 u u ,

, corresponding to

g u

being

function value on decomposition boundary. In case, solving problem (1) is implemented by

following domain decomposition scheme:

1. A given

(0)

g

specified on

.