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Schwarz Method

Overlapping - Dirichlet interface condition
- Jacobin Schwarz Method(JSM): only locally solve the subdomain problem iteratively.
- Restricted additive Schwarz(RAS): First solve the subdomain problem locally, then merge the local solutions by partition of unity.
- Additive Schwarz Method(ASM): First solve the subdomain problem locally, then add the local solutions directly, without using partition of unity.
Nonoverlapping (or overlapping) - Other interface condition (Robin,Fourier)
- P.L.Lions’ Algorithm.
- Després’ Algorithm.
- Optimized Schwarz Method(OSM): Optimize the transmission condition to minimize the iteration number for convergence.

Classical Continuous Schwarz Method

Jacobi Schwarz Algorithm

Jacobi-Schwarz Algorithm (Laplace Equation)
Problem: Solve  on  with  on .
Input: Domain ,  on , 
        initial guesses  on  and  on , 
        boundary values on  and .
Output: Approximation  on .
Initialization: 
            Set  and  as initial guesses in their respective subdomains.
While convergence criterion is not satisfied:

            Solve  on  with:

            Boundary conditions:  on ,  on .
        

            Solve  on  with:

            Boundary conditions:  on ,  on .
        
End While
Return: Combined solution  on .

Restricted Additive Schwarz (RAS)

RAS Algorithm at the Continuous Level (Laplace Equation)
Problem: Solve  on  with  on .
Input: Domain ,  on , 
        initial guesses  on  and  on , 
        boundary values on  and .
Output: Approximation  on .
Initialization: 
            Set  and  as initial guesses in their respective subdomains.
While convergence criterion is not satisfied:

            Compute the residual 
        

            Find local correction :
            Solve  on  with

            Boundary conditions:  on .
        

            Find local correction :
            Solve  on  with

            Boundary conditions:  on .
        

            Compute an average of the local corrections and update the solution:

            
            where  are partition of unity functions and  are extension operators.
        
End While
Return  on .

Additive Schwarz Method (ASM)

AMS Algorithm at the Continuous Level (Laplace Equation)
Problem: Solve  on  with  on .
Input: Domain ,  on , 
        initial guesses  on  and  on , 
        boundary values on  and .
Output: Approximation  on .
Initialization: 
            Set  and  as initial guesses in their respective subdomains.
While convergence criterion is not satisfied:

            Compute the residual 
        

            Find local correction :
            Solve  on  with

            Boundary conditions:  on .
        

            Find local correction :
            Solve  on  with

            Boundary conditions:  on .
        

            Add the local corrections and update the solution directly:

            
            where  are extension operators without using partition of unity functions.
        
End While
Return  on .

Optimized Schwarz Method (OSM)

P.L. Lion’s Algorithm

P.L. Lion's Algorithm (Laplace Equation)
Problem: Solve  on  with  on .
Input: Domain ,  on , 
        initial guesses  on  and  on , 
        boundary values on  and .
Output: Approximation  on .
Initialization: 
            Set  and  as initial guesses in their respective subdomains.
While convergence criterion is not satisfied:

            Solve  on  with 
            
            and  on .
        

            Solve  on  with 
            
            and  on .
        
End While
Return Combined solution  on .

Després’ Algorithm

Després' Algorithm (Helmholtz Equation)
Problem: Solve  on .
Input: Overlapping Domain Decomposition ,  on , 
        initial guesses  on  and  on , 
        boundary values on  and .
Output: Approximation  on .
Initialization: 
            Set  and  as initial guesses in their respective subdomains.
While convergence criterion is not satisfied:

            Solve  on  with ()
            
            and  on .
        

            Solve  on  with ()
            
            and  on .
        
End While
Return Combined solution  on .