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# Parallel Finite Volume Discretization
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The code for this example is located in `examples/FVDiscr`
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## Brief
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This example uses simple two-point FVM scheme to solve Poisson's equation in unit cube domain.
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## Description
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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)_.
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This example may run in both serial and parallel modes with `NP` processes.
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The code loads the mesh for unit cube domain.
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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.
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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.
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Two-point FVM scheme is only valid when cell faces are orthogonal to segments connecting centers of neighboring cells.
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Optionally the code saves the generated matrix and right-hand side in user provided files.
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The distributed matrix is solved using `INNER_ILU2` solver.
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The solution is compared with known exact solution and _C_ and _L₂_ norms are computed.
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The result mesh is saved either in `result.vtk`, or `result.pvtk` depending on number of NP.
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## Arguments
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```
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Usage: ./FVDiscr mesh_file [A.mtx b.rhs]
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```
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- First parameter is the mesh file.
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- Two optional parameters – output file names for generated matrix and right-hand side.
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## Running example
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If you compiled INMOST with `USE_PARTITIONER=OFF` you should provide the prepartitioned mesh, otherwise you can provide either serial mesh, or prepartitioned mesh.
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You can generate meshes using [[GridGen|1390-GridGen-Example]] generator. The following line uses `/tmp/grid-32-32-32.pvtk` mesh from [[GridGen example|1390-GridGen-Example#running-example]].
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```
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$ cd examples/FVDiscr
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$ mpirun -np 4 ./FVDiscr /tmp/grid-32-32-32.pvtk /tmp/A.mtx /tmp/b.rhs
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./FVDiscr
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Processors: 4
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Load(MPI_File): 0.34929
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Assign id: 0.00495315
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Exchange ghost: 0.0186219
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Matrix assemble: 0.232133
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Save matrix "/tmp/A.mtx" and RHS "/tmp/b.rhs": 0.368458
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Solve system: 0.1323542e-06 | 1e-05
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err_C = 0.00237783
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err_L2 = 0.000736251
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Compute true residual: 0.561696
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Retrieve data: 0.000573874
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Exchange phi: 5.00679e-06
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Save "result.pvtk": 0.150515
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```
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If you have ParaView installed, you can open the result mesh file:
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```
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$ paraview --data=result.pvtk
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```
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You can view the following tags:
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- `Solution` – the solution to the problem
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- `K` – tensor K (constant equal to 1 in this example) |
