Parallel Finite Volume Discretization

The code for this example is located in examples/FVDiscr

Brief

This example uses simple two-point FVM scheme to solve Poisson's equation in unit cube domain.

Description

This examples is used to solve the problem div(K grad U) = f with Dirichlet boundary conditions, where K is unit tensor and the right-hand side f is computed from the exact solution: U = sin(PI·x)·sin(PI·y)·sin(PI·z).

This example may run in both serial and parallel modes with NP processes.

The code loads the mesh for unit cube domain. If INMOST is built with USE_PARTITIONER=ON and input mesh is a serial mesh, then the Inner_RCM partitioner is used to partition the mesh.

One layer of ghost cells is created and exchanged. The simplest two-point FVM scheme is used to assemble local matrices. Using ghost cells effectively links local matrices in global matrix. Two-point FVM scheme is only valid when cell faces are orthogonal to segments connecting centers of neighboring cells.

Optionally the code saves the generated matrix and right-hand side in user provided files. The distributed matrix is solved using INNER_ILU2 solver. The solution is compared with known exact solution and C and L₂ norms are computed. The result mesh is saved either in result.vtk, or result.pvtk depending on number of NP.

Arguments

Usage: ./FVDiscr mesh_file [A.mtx b.rhs]

• First parameter is the mesh file.
• Two optional parameters – output file names for generated matrix and right-hand side.

Running example

If you compiled INMOST with USE_PARTITIONER=OFF you should provide the prepartitioned mesh, otherwise you can provide either serial mesh, or prepartitioned mesh.

GridGen/tmp/grid-32-32-32.pvtkGridGen example

$ cd examples/FVDiscr
$ mpirun -np 4 ./FVDiscr /tmp/grid-32-32-32.pvtk /tmp/A.mtx /tmp/b.rhs                                                                   
./FVDiscr                                                                                                                                                                                            
Processors: 4                                                                                                                                                                                        
Load(MPI_File): 0.34929
Assign id: 0.00495315
Exchange ghost: 0.0186219
Matrix assemble: 0.232133
Save matrix "/tmp/A.mtx" and RHS "/tmp/b.rhs": 0.368458
Solve system: 0.1323542e-06 | 1e-05
err_C  = 0.00237783
err_L2 = 0.000736251
Compute true residual: 0.561696
Retrieve data: 0.000573874
Exchange phi: 5.00679e-06
Save "result.pvtk": 0.150515

If you have ParaView installed, you can open the result mesh file:

$ paraview --data=result.pvtk

You can view the following tags:

• Solution – the solution to the problem
• K – tensor K (constant equal to 1 in this example)