Heat equation with PETSc : How can the global solution be obtained from the solutions on the subdomains?

General Discussion
1

Hi FreeFem community,

I am currently using PETSc to solve the transient heat equation (DMeshCreate), but at each time step I need to compute cost functionals, which require the global solution. How can I obtain the global solution from the local solutions?

Additionally, I would like to know how to save this global solution for visualization in ParaView.

Thank you in advance for your answers.

2

I doubt that’s true.

How can I obtain the global solution from the local solutions?

Search for N2O in the examples/hpddm folder.

3

When you say you want to compute a cost functional, probably it implies that you have to compute an integral over the whole domain. There is a standard way to do it: compute each integral in each subdomain, and add up all of them. However, doing that you have to introduce a partition of unity in order to avoid taking twice the integral on the overlapping regions.
The typical code lines for that are

//build partition of unity
fespace Vhpart(Th, P0);  // P0 elements for partition function
Vhpart part;  // Partition function
PartitionCreate(Th, part[], P0);  // Create partition using the mesh Th, part[] takes values 0 or 1

real localint = int2d(Th)(part*sol);//integral over the subdomain, without overlapping
real globalint;//global integral (has to be computed)
mpiAllReduce(localint, globalint, mpiCommWorld, mpiSUM);//sum up the local integrals to get the global integral
if (mpirank==0) cout << "The global integral is " << globalint << endl;

About visualization with paraview, you can do as (for example)

fespace V1(Th,P1);
V1 sol;// unknown in P1
sol=(x<0.5)? 0. : 1.;
int[int] myorder=[1];
savevtk("sol.vtu",Th,sol,order=myorder,bin=1);

It will create several files: sol_nbproc.pvd where nbproc is the number of process, and the files sol_nbproc_…vtu, one for each proc.
The file you have to load with paraview is the master file sol_nbproc.pvd (it will load all the proc files together).