Once you have developed a phase field model using implicit time integration, you need to solve your system of nonlinear equations using MOOSE. MOOSE uses the PETSc package to solve the equations, therefore PETSc options that we have found to work well for phase field modeling are provided here.

## Solution Methods

Three solve methods are available in MOOSE, which are set in the Executioner block using the solve_type parameter, where the three possible options are:

• NEWTON - Direct solution of the system of equations using Newton's method. The full and accurate Jacobian is required. Thus, the Preconditioner block must be employed for systems with multiple nonlinear variables.
• JFNK - The system is solved using Jacobian Free Newton Krylov (JFNK), so no Jacobian terms are needed. However, JFNK often does not perform well without preconditioning.
• PJFNK - The system is solved using preconditioned JFNK. The Jacobian is used to precondition the matrix, but it does not have to be fully correct and can neglect off-diagonal terms.

## Preconditioning Options

When using NEWTON or PJFNK, PETSc is used to manipulate the Jacobian/preconditioner matrix. The method used for this manipulation is defined using PETSc options in the Executioner block. Determining the optimal approach can be difficult, but we have identified certain options that work well for phase field problems. They are summarized below:

### LU

This uses LU decomposition to directly apply the inverse of the Jacobian matrix. It is typically the most accurate approach, but is expensive and only works on one processor. It is often a good debugging tool. Typical options in the block are

petsc_options_iname = '-pc_type'
petsc_options_value = 'lu'


### ASM

The Additive Schwartz Method (ASM), is a domain decomposition method. It works well for most models, and is the only method that works well with the split Cahn-Hilliard equations. Increasing the -pc_asm_overlap improves performance but increases the computational expense. Typical options are

petsc_options_iname = '-pc_type -ksp_grmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
petsc_options_value = 'asm      31                  preonly       lu           2'


#### ASM/ILU

The incomplete factorization method (ILU) allows specifying a fill factor using -pc_factor_levels increasing this can improve the preconditioner.

### BoomerAMG

BoomerAMG is the Hypre implementation of the Algebraic MultiGrid Method. It works well for solving the Allen-Cahn equation and the direct solution of the Cahn-Hilliard method. It has horrible performance with solving the split Cahn-Hilliard equations. Common options are

petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart -pc_hypre_boomeramg_strong_threshold'
petsc_options_value = 'hypre    boomeramg      31                 0.7'