Additive Schwarz Method - A Technical Example

A Technical Example

Here we assume that the reader is familiar with partial differential equations.

We will be solving the partial differential equation

u_xx + u_yy = f (**)

The boundary condition is boundedness at infinity.

We decompose the domain R² into two overlapping subdomains H₁ = (− ∞,1] × R and H₂ = [0,+ ∞) × R. In each subdomain, we will be solving a BVP of the form:

u( j )_xx + u( j )_yy = f in H_j

u( j )(x_j,y) = g(y)

where x₁ = 1 and x₂ = 0 and taking boundedness at infinity as the other boundary condition. We denote the solution u( j ) of the above problem by S(f,g). Note that S is bilinear.

The Schwarz algorithm proceeds as follows:

1. Start with approximate solutions u( 1 )₀ and u( 2 )₀ of the PDE in subdomains H₁ and H₂ respectively. Initialize k to 1.
2. Calculate u( j )_{k + 1} = S(f,u(3 − j)_k(x_j)) with j = 1,2.
3. Increase k by one and repeat 2 until sufficient precision is achieved.