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SNES

The SNES (Scalable Nonlinear Equations Solvers) module provides methods for solving nonlinear systems of the form F(x) = 0. It builds on KSP for the linear solves within Newton-like methods.

Overview

SNES provides:

  • Newton methods: Newton line search, Newton trust region
  • Quasi-Newton: L-BFGS, Broyden
  • Nonlinear Richardson: With various line search strategies
  • FAS multigrid: Full Approximation Scheme for nonlinear problems
  • Composite solvers: Combine multiple nonlinear solvers

Creating a SNES Solver

# Basic creation
snes = SNES(petsclib, MPI.COMM_WORLD)

# With options
snes = SNES(petsclib, MPI.COMM_WORLD;
    snes_type = "newtonls",
    snes_rtol = 1e-8,
    snes_max_it = 50
)

Setting the Nonlinear Function

Define the residual function F(x):

function residual!(fx, snes, x)
    # Compute F(x) and store in fx
    fx[1] = x[1]^2 + x[2] - 1
    fx[2] = x[1] + x[2]^2 - 1
    return 0
end

setfunction!(snes, residual!, f_vec)

Setting the Jacobian

Define the Jacobian J = dF/dx:

function jacobian!(J, snes, x)
    # Fill Jacobian matrix
    J[1, 1] = 2*x[1]
    J[1, 2] = 1.0
    J[2, 1] = 1.0
    J[2, 2] = 2*x[2]
    assemble!(J)
    return 0
end

setjacobian!(snes, jacobian!, J, J)  # (J, P) where P is preconditioner matrix

Using a DM

For PDE problems, associate the SNES with a DM:

setDM!(snes, dm)

# Get the DM from SNES
dm = getDMDA(snes)

Solving

# Solve with initial guess x
solve!(x, snes)

# Get solution vector
sol = get_solution(snes)

Common Solver Options

Nonlinear Solver Types (snes_type)

  • newtonls - Newton with line search (default)
  • newtontr - Newton with trust region
  • nrichardson - Nonlinear Richardson
  • qn - Quasi-Newton (L-BFGS)
  • fas - Full Approximation Scheme multigrid

Convergence Options

  • snes_rtol - Relative tolerance
  • snes_atol - Absolute tolerance
  • snes_stol - Step tolerance
  • snes_max_it - Maximum iterations
  • snes_monitor - Print residual each iteration

Line Search Options

  • snes_linesearch_type - bt (backtracking), basic, l2, cp

Example: Full Setup

petsclib = PETSc.petsclibs[1]
PETSc.initialize(petsclib)

snes = SNES(petsclib, MPI.COMM_WORLD;
    snes_monitor = true,
    ksp_type = "gmres",
    pc_type = "ilu"
)

setfunction!(snes, residual!, f)
setjacobian!(snes, jacobian!, J, J)
setfromoptions!(snes)

solve!(x, snes)

Functions

PETSc.SNES — Method
SNES(petsclib, comm::MPI.Comm; prefix="", options...)

Create a PETSc nonlinear solver (SNES) context on the communicator comm.

Arguments

  • petsclib: The PETSc library instance
  • comm::MPI.Comm: MPI communicator
  • prefix::String: Optional prefix for command-line options
  • options...: Additional PETSc options as keyword arguments

If comm has size 1, the garbage collector will handle cleanup automatically. Otherwise, the user is responsible for calling destroy.

External Links

source
PETSc.destroy — Method
destroy(snes::AbstractPetscSNES)

Destroy a SNES (nonlinear solver) object and release associated resources.

This function is typically called automatically via finalizers when the object is garbage collected, but can be called explicitly to free resources immediately.

External Links

source
PETSc.setfunction! — Method
setfunction!(snes, f!, vec)
setfunction!(f!, snes, vec)

Set the residual function f! for the nonlinear solver snes.

The function f! will be called as f!(fx, snes, x) where:

  • fx: Output vector to store the residual F(x)
  • snes: The SNES context
  • x: Input vector with current solution

The vec argument is a template vector used for the residual.

External Links

source
PETSc.setjacobian! — Function
setjacobian!(
    snes::AbstractSNES,
    updateJ!::Function,
    J::AbstractMat,
    P::AbstractMat = J
)
setjacobian!(
    updateJ!::Function,
    snes::AbstractSNES,
    J::AbstractMat,
    P::AbstractMat = J
)

Define updateJ! to be the function that updates the Jacobian of the snes.

If J == P then a call to updateJ!(J, snes, x) should set the elements of the PETSc Jacobian (approximation).

If J ≠ P then a call to updateJ!(J, P, snes, x) should set the elements of the PETSc Jacobian (approximation) and preconditioning matrix P.

If you set snes.user_ctx, then updateJ! may optionally accept that as an additional last argument:

  • updateJ!(J, snes, x, user_ctx) when J == P
  • updateJ!(J, P, snes, x, user_ctx) when J ≠ P

External Links

source