# [petsc-users] Convergence issues for SNES NASM

Tue May 10 14:12:38 CDT 2022

```Hello,

We are working on a finite difference solver for a 2D nonlinear PDE with
Dirichlet Boundary conditions on a rectangular domain. Our goal is to solve
the problem with parallel nonlinear additive Schwarz (NASM) as the outer
solver. Our code is similar to SNES example 5
<https://petsc.org/release/src/snes/tutorials/ex5.c.html>. In example 5,
the parallel NASM can be executed with a command like `mpiexec -n 4 ./ex5
-mms 3 -snes_type nasm -snes_nasm_type restrict -da_overlap 2` which gives
a convergent result. We assume this is the correct usage. A comment in the
source code for NASM mentions that NASM should be a preconditioner but
there's no documentation on the usage. The Brune paper does not cover
parallel NASM either. We observed that increasing the overlap leads to
fewer Schwarz iterations. The parallelization works seamlessly for an
arbitrary number of subdomains. This is the type of behavior we were
expecting from our code.

Our method uses box-style stencil width d = ceil(N^(1/3)) on a N by N DMDA.
The finite difference stencil consists of 4d+1 points spread out in a
diamond formation. If a stencil point is out of bounds, then it is
projected onto the boundary curve. Since the nodes on the boundary curve
would result in an irregular mesh, we chose not treat boundary nodes as
unknowns as in Example 5. We use DMDACreate2d to create the DA for the
interior points and DMDASNESSetFunctionLocal to associate the residue
function to the SNES object.

Our code works serially. We have also tested our code
with Newton-Krylov-Schwarz (NKS) by running something akin to `mpiexec -n
<n> ./solve -snes_type newtonls`. We have tested the NKS for several
quantities of subdomains and overlap and the code works as expected. We
have some confidence in the correctness of our code. The overlapping NASM
was implemented in MATLAB so we know the method converges. However, the
parallel NASM will not converge with our PETSc code. We don't understand
why NKS works while NASM does not. The F-norm residue monotonically
decreases and then stagnates.

Here is an example of the output when attempting to run NASM in parallel:
takahashi at ubuntu:~/Desktop/MA-DDM/Cpp/Rectangle\$ mpiexec -n 4 ./test1 -t1_N
20 -snes_max_it 50 -snes_monitor -snes_view -da_overlap 3 -snes_type nasm
-snes_nasm_type restrict
0 SNES Function norm 7.244681057908e+02
1 SNES Function norm 1.237688062971e+02
2 SNES Function norm 1.068926073552e+02
3 SNES Function norm 1.027563237834e+02
4 SNES Function norm 1.022184806736e+02
5 SNES Function norm 1.020818227640e+02
6 SNES Function norm 1.020325629121e+02
7 SNES Function norm 1.020149036595e+02
8 SNES Function norm 1.020088110545e+02
9 SNES Function norm 1.020067198030e+02
10 SNES Function norm 1.020060034469e+02
11 SNES Function norm 1.020057582380e+02
12 SNES Function norm 1.020056743241e+02
13 SNES Function norm 1.020056456101e+02
14 SNES Function norm 1.020056357849e+02
15 SNES Function norm 1.020056324231e+02
16 SNES Function norm 1.020056312727e+02
17 SNES Function norm 1.020056308791e+02
18 SNES Function norm 1.020056307444e+02
19 SNES Function norm 1.020056306983e+02
20 SNES Function norm 1.020056306826e+02
21 SNES Function norm 1.020056306772e+02
22 SNES Function norm 1.020056306753e+02
23 SNES Function norm 1.020056306747e+02
24 SNES Function norm 1.020056306745e+02
25 SNES Function norm 1.020056306744e+02
26 SNES Function norm 1.020056306744e+02
27 SNES Function norm 1.020056306744e+02
28 SNES Function norm 1.020056306744e+02
29 SNES Function norm 1.020056306744e+02
30 SNES Function norm 1.020056306744e+02
31 SNES Function norm 1.020056306744e+02
32 SNES Function norm 1.020056306744e+02
33 SNES Function norm 1.020056306744e+02
34 SNES Function norm 1.020056306744e+02
35 SNES Function norm 1.020056306744e+02
36 SNES Function norm 1.020056306744e+02
37 SNES Function norm 1.020056306744e+02
38 SNES Function norm 1.020056306744e+02
39 SNES Function norm 1.020056306744e+02
40 SNES Function norm 1.020056306744e+02
41 SNES Function norm 1.020056306744e+02
42 SNES Function norm 1.020056306744e+02
43 SNES Function norm 1.020056306744e+02
44 SNES Function norm 1.020056306744e+02
45 SNES Function norm 1.020056306744e+02
46 SNES Function norm 1.020056306744e+02
47 SNES Function norm 1.020056306744e+02
48 SNES Function norm 1.020056306744e+02
49 SNES Function norm 1.020056306744e+02
50 SNES Function norm 1.020056306744e+02
SNES Object: 4 MPI processes
type: nasm
total subdomain blocks = 4
Local solver information for first block on rank 0:
Use -snes_view ::ascii_info_detail to display information for all blocks
SNES Object: (sub_) 1 MPI processes
type: newtonls
maximum iterations=50, maximum function evaluations=10000
tolerances: relative=1e-08, absolute=1e-50, solution=1e-08
total number of linear solver iterations=22
total number of function evaluations=40
norm schedule ALWAYS
Jacobian is built using a DMDA local Jacobian
SNESLineSearch Object: (sub_) 1 MPI processes
type: bt
interpolation: cubic
alpha=1.000000e-04
maxstep=1.000000e+08, minlambda=1.000000e-12
tolerances: relative=1.000000e-08, absolute=1.000000e-15,
lambda=1.000000e-08
maximum iterations=40
KSP Object: (sub_) 1 MPI processes
type: preonly
maximum iterations=10000, initial guess is zero
tolerances:  relative=1e-05, absolute=1e-50, divergence=10000.
left preconditioning
using NONE norm type for convergence test
PC Object: (sub_) 1 MPI processes
type: lu
out-of-place factorization
tolerance for zero pivot 2.22045e-14
matrix ordering: nd
factor fill ratio given 5., needed 2.13732
Factored matrix follows:
Mat Object: 1 MPI processes
type: seqaij
rows=169, cols=169
package used to perform factorization: petsc
total: nonzeros=13339, allocated nonzeros=13339
using I-node routines: found 104 nodes, limit used is 5
linear system matrix = precond matrix:
Mat Object: 1 MPI processes
type: seqaij
rows=169, cols=169
total: nonzeros=6241, allocated nonzeros=6241
total number of mallocs used during MatSetValues calls=0
not using I-node routines
maximum iterations=50, maximum function evaluations=10000
tolerances: relative=1e-08, absolute=1e-50, solution=1e-08
total number of function evaluations=51
norm schedule ALWAYS
Jacobian is built using a DMDA local Jacobian
problem ex10 on 20 x 20 point 2D grid with d = 3, and eps = 0.082:
error |u-uexact|_inf = 3.996e-01, |u-uexact|_h = 2.837e-01

We have been stuck on this for a while now. We do not know how to debug
this issue. Please let us know if you have any insights.
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