PETSc

While Julia has quite a lot of linear algebra capability built-in, it is still missing a few things. In particular, Julia does not contain built-in iterative linear solvers (like Krylov solvers). Even though there are some third-party Krylov solvers (such KrylovMethods.jl ) they lack preconditioning and parallelism.

For this reason, it is useful to utilize the PETSc library from Argonne National Laboratory. PETSc is a mature code-base providing parallel Krylov solvers and preconditioners that are a perfect match for FEM. To use them, I have developed a Julia wrapper: MiniPETSc.jl which MOOSE.jl takes advantage of.

PETSc Setup

Before using PETSc it's critical that you've been through the setup steps on the Installation page. PETSc needs to be compiled against your MPI installation, installed and the PETSC_DIR environment variable needs to be set to point to that installation.

Using PETSc

To use PETSc within your MOOSE.jl script... all that is necessary is to switch the type of solver you are using to one of the Petsc* solvers. For instance, here is a modified form of the script from Example #2:

using MOOSE

# Create the Mesh
mesh = buildSquare(0, 1, 0, 1, 10, 10)

# Create the System to hold the equations
diffusion_system = System{Dual{4,Float64}}(mesh)

# Add a variable to solve for
u = addVariable!(diffusion_system, "u")

# Apply the Laplacian operator to the variable
addKernel!(diffusion_system, Diffusion(u))

# u = 0 on the Left
addBC!(diffusion_system, DirichletBC(u, [4], 0.0))

# u = 1 on the Right
addBC!(diffusion_system, DirichletBC(u, [2], 1.0))

# Initialize the system of equations
initialize!(diffusion_system)

# Create a solver and solve
solver = PetscImplicitSolver(diffusion_system)
solve!(solver)

# Output
out = VTKOutput()
output(out, solver, "simple_diffusion_out")

The PetscImplicitSolver will utilize PETSc vectors, matrices and solvers to solve the system.

PETSc Options

By default PETSc will use a GMRES solver with ILU-0 preconditioning. For small, simple problems this will work fine. But: for anything more complicated you will want to specify options to PETSc to change the solver/preconditioner. This can be achieved using the normal PETSc command-line syntax.

For instance, to use the algebraic multigrid Hypre/Boomeramg package for preconditioning you would run your script like so:

julia myscript.jl -pc_type hypre -pc_hypre_type boomeramg

To see a lot of the command-line options you can use try running your script using the -help option:

julia myscript.jl -help

Here are some of the most useful/common PETSc options:

Name	Description
-ksp_monitor	View convergence information
-snes_ksp_ew	Variable linear solve tolerance, useful for transient solves
-help	Show PETSc options during the solve

Name	Value	Description
-pc-type	ilu	Default for serial
	bjacobi	Default for parallel with -sub_pc_type ilu
	asm	Additive Schwartz with -sub_pc_type ilu
	lu	Full LU, serial only
	gamg	Generalized Geometric-Algebric MultiGrid
	hypre	Hypre, usually used with boomeramg
-sub_pc_type	ilu, lu, hypre	Can be used with bjacobi or asm
-pc_hypre_type	boomeramg	Algebraic Multigrid
-ksp_gmres_restart	# (default = 30)	Number of Krylov vectors to store

For even more, refer to the PETSc documentation