help with a problem

[Andre17420]

Hello to everybody,
I am a student from Milan and I have just started facing numerical methods. I gotta solve this parabolic problem, but I'm stuck in checking the stability and the convergency. I attached two pictures of the problem. Thank you all.
Here is what I have done so far:

Define the mesh

n = 20
mesh = UnitSquareMesh(n, n, diagonal='crossed')

Define the function space

V = FunctionSpace(mesh, 'Lagrange', 1)

Boundary condition

g = Constant(1.0)
bc = DirichletBC(V, g, "on_boundary")

Problem

x = SpatialCoordinate(mesh)
f_fun = lambda t: (pi/2cos(pit) + pi**2sin(pit)) * sin(pix[0]) * sin(pix[1])

T = 1.
dt = 0.1
theta = 0.5

Exact solution

u_ex = lambda t: 1 + 0.5 * sin(pit) * sin(pix[0]) * sin(pi*x[1])

Variational formulation

u = TrialFunction(V)
v = TestFunction(V)

t = 0
u_old = interpolate(u_ex(t), V)

a = 1/dt * u * v * dx + theta * dot(grad(u), grad(v)) * dx

Problem definition

u_h = Function(V, name='Temperature')
u_h.assign(u_old)

start_time = perf_counter()
A = assemble(a, bcs=bc)
solver = LinearSolver(A,
solver_parameters={"ksp_type": "preonly",
"pc_type": "lu"})

Time stepping

errL2 = 0
errH1 = 0

while t+dt/2<T:
t_old = t
t += dt
print('t = ', t)

L = ( 1/dt * u_old * v * dx - (1-theta) * (dot(grad(u_old), grad(v)) * dx)
     + theta * f_fun(t) * v * dx + (1-theta) * (f_fun(t_old) * v * dx) )
b = assemble(L, bcs=bc)
solver.solve(u_h, b)
u_old = u_h

#outfile.write(u_h, time=t)

# Compute the error
errL2 = max(errL2, errornorm(u_ex(t), u_h, 'L2'))
errH1 = max(errH1, errornorm(u_ex(t), u_h, 'H1'))

exec_time = perf_counter() - start_time

[colinjcotter]

Hello Andre, What is your question? all the best cjc On 9 Apr 2023, at 11:22, Andre17420 ***@***.***> wrote:  Hello to everybody, I am a student from Milan and I have just started facing numerical methods. I gotta solve this parabolic problem, but I'm stuck in checking the stability and the convergency. I attached two pictures of the problem. Thank you all. Here is what I have done so far: Define the mesh n = 20 mesh = UnitSquareMesh(n, n, diagonal='crossed') Define the function space V = FunctionSpace(mesh, 'Lagrange', 1) Boundary condition g = Constant(1.0) bc = DirichletBC(V, g, "on_boundary") Problem x = SpatialCoordinate(mesh) f_fun = lambda t: (pi/2cos(pit) + pi**2sin(pit)) * sin(pix[0]) * sin(pix[1]) T = 1. dt = 0.1 theta = 0.5 Exact solution u_ex = lambda t: 1 + 0.5 * sin(pit) * sin(pix[0]) * sin(pi*x[1]) Variational formulation u = TrialFunction(V) v = TestFunction(V) t = 0 u_old = interpolate(u_ex(t), V) a = 1/dt * u * v * dx + theta * dot(grad(u), grad(v)) * dx Problem definition u_h = Function(V, name='Temperature') u_h.assign(u_old) start_time = perf_counter() A = assemble(a, bcs=bc) solver = LinearSolver(A, solver_parameters={"ksp_type": "preonly", "pc_type": "lu"}) Time stepping errL2 = 0 errH1 = 0 while t+dt/2<T: t_old = t t += dt print('t = ', t) L = ( 1/dt * u_old * v * dx - (1-theta) * (dot(grad(u_old), grad(v)) * dx) + theta * f_fun(t) * v * dx + (1-theta) * (f_fun(t_old) * v * dx) ) b = assemble(L, bcs=bc) solver.solve(u_h, b) u_old = u_h #outfile.write(u_h, time=t) # Compute the error errL2 = max(errL2, errornorm(u_ex(t), u_h, 'L2')) errH1 = max(errH1, errornorm(u_ex(t), u_h, 'H1')) exec_time = perf_counter() - start_time [g1]<https://user-images.githubusercontent.com/130237366/230764888-bc73995f-3cac-4745-99f9-a10f70958752.png> [image]<https://user-images.githubusercontent.com/130237366/230764921-40ebe7f8-75a2-4046-9bde-45452e9cf8a3.png> — Reply to this email directly, view it on GitHub<#2867>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/ABOSV4QOAQ6ZJS5HO7IX7F3XAJ5T5ANCNFSM6AAAAAAWX75RMU>. You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[Andre17420]

I'm not able to check the stability and the convergence, I'm asking wheter anybody could help me out

[colinjcotter]

It sounds like this is a conceptual problem rather than a Firedrake one. I think that the best approach to learning this would be to look at the definitions of stability and convergence that you have been provided with and think about what you would need to measure in order to check if they are satisfied. On 9 Apr 2023, at 10:56, Andre17420 ***@***.***> wrote:  I'm not able to check the stability and the convergence, I'm asking wheter anybody could help me out — Reply to this email directly, view it on GitHub<#2867 (reply in thread)>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/ABOSV4XGDZ2FCQS5KHLNKYTXAKBVVANCNFSM6AAAAAAWX75RMU>. You are receiving this because you commented.Message ID: ***@***.***>

[Andre17420]

can you help me?