# Space-time domain decomposition preconditioned GMRES algorithm for linear parabolic equations

## Description

We implement the multilevel space-time pure multiplicative Schwarz (STPMS) algorithm to solve the linear parabolic equations, i.e., the linear parabolic equations are solved by the multilevel multiplicative Schwarz preconditioned GMRES method in parallel on both space and time dimensions. The data shows the optimality and scalability of STPMS algorithm, i.e., it shows that the iteration counts of STPMS decrease with the increase of the overlap “ovlp”, and are bounded independently of the window size “s” and the number of subdomains “N_p”.

## Files

## Steps to reproduce

Some notes on the code: (1) mg_method='multiplicative' + inner_pc='asm' denotes STPMS algorithm in our paper (2) mg_method='multiplicative' + inner_pc='msm' denotes STMS algorithm in our paper (3) excase=1 and excase=2 denotes example 1 and example 2 in our paper; (4) M, N, P denote the number of mesh nodes in x, y, z direction, respectively (5) m, n, p denote number of processes in x, y, z direction, respectively (6) ovlp deonte thes overlapping size in x, y, z direction, respectively The code can be reproduced by the following settings and obtain the date. (1) For STMS, choosing mg_method='multiplicative' and inner_pc='asm'. Setting excase=1, M=129; Varying the number of subdomains (k=1:3), the window size (i=5:9), the overlapping size (j=1:3) (2) For STPMS, choosing mg_method='multiplicative' and inner_pc='msm'. Setting excase=1, M=129; Varying the number of subdomains (k=1:3), the window size (i=5:9), the overlapping size (j=1:3)