There's an algorithm called ADI (Alternating Direction Implicit) in applied math circles and Split-operator in physics circles that does basically what you describe. It's an iterative method, and it follows this basic procedure:

1. For every value of $y$ , relax in the $x$-direction. This matrix should be tridiagonal, so it can be solved exactly in relatively little time.

2. For every value of $x$ , relax in the $y$-direction. Again, this should be pretty quick.

3. Repeat 1 and 2 until the error is as small as you want it to be.

I don't know the formal complexity of this algorithm, but I've found it to converge in fewer iterations than things like Jacobi and Gauss-Seidel every time I've used it.

There's an algorithm called ADI (Alternating Direction Implicit) in applied math circles and Split-operator in physics circles that does basically what you describe. It's an iterative method, and it follows this basic procedure:

1. For every value of $y$ , relax in the $x$-direction. This matrix should be tridiagonal, so it can be solved directly in relatively little time.

2. For every value of $x$ , relax in the $y$-direction. Again, this should be pretty quick.

3. Repeat 1 and 2 until the error is as small as you want it to be.

I don't know the formal complexity of this algorithm, but I've found it to converge in fewer iterations than things like Jacobi and Gauss-Seidel every time I've used it.