# Using different time steps in different subdomains

**URL:** <https://fenicsproject.discourse.group/t/using-different-time-steps-in-different-subdomains/9085>\
**Category:** variational formulation\
**Created:** [August 22, 2022, 3:27pm UTC](https://fenicsproject.discourse.group/t/using-different-time-steps-in-different-subdomains/9085 "2022-08-22T15:27:41Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![r\_s](https://avatars.discourse-cdn.com/v4/letter/r/a88e57/32.png) [@r\_s](https://fenicsproject.discourse.group/u/r_s)\
**Post date:** [August 22, 2022, 3:27pm UTC](https://fenicsproject.discourse.group/t/using-different-time-steps-in-different-subdomains/9085/1 "2022-08-22T15:27:41Z")

</div>

Hello,

I want to implement a time-dependent problem where different time step sizes are used in subdomains. I use FEniCSx 0.4.1 but an update to the latest version would be no problem.

To illustrate what I want to achieve, consider the given picture:

 ![image](https://global.discourse-cdn.com/free1/uploads/fenicsproject1/original/2X/3/31c7a2d61b5b085b4b3e851bbb8d24755d7d3232.png)

I have a domain consisting of two subdomains. I know that due to material parameters the solution in subdomain 1 changes more within a certain period of time than in subdomain 2 (which can be considered constant within that time period). Therefore, I want to only calculate the solution in subdomain 1 for N time steps, update subdomain 2 after N time steps, and repeat. To implement this approach, I need a way to somehow fixate the solution in subdomain 2 or apply a known solution.

While searching for similar problems, I encountered this: [Integrating on subdomain](https://fenicsproject.discourse.group/t/integrating-on-subdomain/5084). But all answers there and for similar questions are given for legacy FEniCS and I struggle to apply them to FEniCSx.

As a simple model problem, I modified the heat equation form the tutorial. Now 2 different materials are used and only subdomain 1 should be calculated. The entire code is given at the end.

I came up with the following idea (assuming the subdomains are marked as 1 and 2):

```auto
a = u * v * dx(1) + dt*ufl.dot(ufl.grad(u), ufl.grad(v)) * dx(1) 
L = (u_n + dt * f) * v * dx(1)
a += u_known * dx(2)
L += u_known * dx(2)

```

Where I try to use a known solution in subdomain 2. Obviously the last two lines are wrong. My questions are: How can I apply a known solution in subdomain 2? Is there a different approach that I should use? Is what I’m trying to do even possible in FEniCSx?

Thank you in advance!

```auto
import numpy as np
from mpi4py import MPI
from petsc4py import PETSc
from petsc4py.PETSc import ScalarType
from dolfinx import fem, mesh, io, plot

# Define temporal parameters
t = 0 # Start time
T = 2.0 # Final time
num_steps = 61     
dt = T / num_steps # time step size

# Define mesh
nx, ny = 50, 50
domain = mesh.create_rectangle(MPI.COMM_WORLD, [np.array([-2, -2]), np.array([2, 2])], [nx, ny], mesh.CellType.triangle)
V = fem.FunctionSpace(domain, ("CG", 1))

Q = fem.FunctionSpace(domain, ("DG", 0))

def Omega_1(x):
    return x[1] >= 0.0

def Omega_2(x):
    return x[1] <= 0.0

kappa = fem.Function(Q)
cells_1 = mesh.locate_entities(domain, domain.topology.dim, Omega_1)
cells_2 = mesh.locate_entities(domain, domain.topology.dim, Omega_2)

kappa.x.array[cells_1] = np.full_like(cells_1, 1, dtype=ScalarType)
kappa.x.array[cells_2] = np.full_like(cells_2, 0.01, dtype=ScalarType)

# Create initial condition
def initial_condition(x, a=5):
    return np.exp(-a*(x[0]**2+x[1]**2))
u_n = fem.Function(V)
u_n.name = "u_n"
u_n.interpolate(initial_condition)

# Create boundary condition
fdim = domain.topology.dim - 1
boundary_facets = mesh.locate_entities_boundary(
    domain, fdim, lambda x: np.full(x.shape[1], True, dtype=bool))
bc = fem.dirichletbc(PETSc.ScalarType(0), fem.locate_dofs_topological(V, fdim, boundary_facets), V)

xdmf = io.XDMFFile(domain.comm, "output.xdmf", "w")
xdmf.write_mesh(domain)

# Define solution variable, and interpolate initial solution for visualization in Paraview
uh = fem.Function(V)
uh.name = "uh"
uh.interpolate(initial_condition)
xdmf.write_function(uh, t)

import ufl
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
f = fem.Constant(domain, PETSc.ScalarType(0))

# The following two lines of code need to be changed in a way that
# only subdomain 1 is calculated and subdomain 2 remains unchanged.
a = u * v * ufl.dx + dt*kappa*ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx 
L = (u_n + dt * f) * v * ufl.dx
##################################################################

bilinear_form = fem.form(a)
linear_form = fem.form(L)

A = fem.petsc.assemble_matrix(bilinear_form, bcs=[bc])
A.assemble()
b = fem.petsc.create_vector(linear_form)

solver = PETSc.KSP().create(domain.comm)
solver.setOperators(A)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)

for i in range(num_steps):
    t += dt

    # Update the right hand side reusing the initial vector
    with b.localForm() as loc_b:
        loc_b.set(0)
    fem.petsc.assemble_vector(b, linear_form)
    
    # Apply Dirichlet boundary condition to the vector
    fem.petsc.apply_lifting(b, [bilinear_form], [[bc]])
    b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
    fem.petsc.set_bc(b, [bc])

    # Solve linear problem
    solver.solve(b, uh.vector)
    uh.x.scatter_forward()

    # Update solution at previous time step (u_n)
    u_n.x.array[:] = uh.x.array

    # Write solution to file
    xdmf.write_function(uh, t)

xdmf.close()

```

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<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [August 23, 2022, 4:42pm UTC](https://fenicsproject.discourse.group/t/using-different-time-steps-in-different-subdomains/9085/2 "2022-08-23T16:42:41Z")

</div>

You would need me to use a Dirichlet boundary condition on all the interior nodes of the subdomain you would like to pin for a set of time steps.

---

<div class="post-metadata">

**Author:** ![r\_s](https://avatars.discourse-cdn.com/v4/letter/r/a88e57/32.png) [@r\_s](https://fenicsproject.discourse.group/u/r_s)\
**Post date:** [August 25, 2022, 2:32pm UTC](https://fenicsproject.discourse.group/t/using-different-time-steps-in-different-subdomains/9085/3 "2022-08-25T14:32:47Z")

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Thank you for your help!