4-th order ODE

[HumbertHumbert7]

I’m trying to solve a simoleODE: u’’’’=1 With BCs u(0)=u(1)=0 and u’’(0)=u’’(1)=0 My code:

import firedrake as fd import numpy as np import matplotlib.pyplot as plt

N=20 mesh=fd.UnitIntervalMesh(N)

V=fd.FunctionSpace(mesh,'CG',2)

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

Def. BC

u_d=fd.Constant(0.) #Dir. BC bc=fd.DirichletBC(V,u_d,'on_boundary')

x=fd.SpatialCoordinate(mesh)[0] #x is a vect., we take the first component. x is used for the Firedrake funct. f=fd.Constant(-1.0)

Weak form.

a=fd.dot(fd.div(fd.grad(u)),fd.div(fd.grad(v)))fd.dx L=fv*fd.dx

u_h=fd.Function(V) fd.solve(a==L,u_h,bc)

fig,ax=plt.subplots() fd.plot(u_h,axes=ax,color='g')

I have an error in the solver. Why?

[HumbertHumbert7]

u’’’’=-1 actually

[colinjcotter]

Your discretisation space is not contained in H^2 as is required for fourth order operators. You would need to use an H^2 conforming space. In 1D, cubic Hermite works.

From: HumbertHumbert7 @.> Date: Friday, 15 March 2024 at 16:35 To: firedrakeproject/firedrake @.> Cc: Subscribed @.***> Subject: [firedrakeproject/firedrake] 4-th order ODE (Issue #3456)

I’m trying to solve a simoleODE: u’’’’=1 With BCs u(0)=u(1)=0 and u’’(0)=u’’(1)=0 My code:

import firedrake as fd import numpy as np import matplotlib.pyplot as plt

N=20 mesh=fd.UnitIntervalMesh(N)

V=fd.FunctionSpace(mesh,'CG',2)

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

Def. BC

u_d=fd.Constant(0.) #Dir. BC bc=fd.DirichletBC(V,u_d,'on_boundary')

x=fd.SpatialCoordinate(mesh)[0] #x is a vect., we take the first component. x is used for the Firedrake funct. f=fd.Constant(-1.0)

Weak form.

a=fd.dot(fd.div(fd.grad(u)),fd.div(fd.grad(v)))fd.dx L=fv*fd.dx

u_h=fd.Function(V) fd.solve(a==L,u_h,bc)

fig,ax=plt.subplots() fd.plot(u_h,axes=ax,color='g')

I have an error in the solver. Why?

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[HumbertHumbert7]

Thank you very much, so what do I have to change?

[colinjcotter]

The finite element space.

From: HumbertHumbert7 @.> Date: Friday, 15 March 2024 at 16:55 To: firedrakeproject/firedrake @.> Cc: Cotter, Colin J @.>, Comment @.> Subject: Re: [firedrakeproject/firedrake] 4-th order ODE (Issue #3456)

Thank you very much, so what do I have to change?

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[HumbertHumbert7]

And how can I set the cubic hermite?

[colinjcotter]

https://www.firedrakeproject.org/variational-problems.html#supported-finite-elements

From: HumbertHumbert7 @.> Date: Friday, 15 March 2024 at 16:57 To: firedrakeproject/firedrake @.> Cc: Cotter, Colin J @.>, Comment @.> Subject: Re: [firedrakeproject/firedrake] 4-th order ODE (Issue #3456)

And how can I set the cubic hermite?

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[HumbertHumbert7]

Thank you, I’ll try

[colinjcotter]

Also note that Dirichlet BCs are not supported for these spaces so you will need to use a Nitche method.

From: HumbertHumbert7 @.> Date: Friday, 15 March 2024 at 16:59 To: firedrakeproject/firedrake @.> Cc: Cotter, Colin J @.>, Comment @.> Subject: Re: [firedrakeproject/firedrake] 4-th order ODE (Issue #3456)

Thank you, I’ll try

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[HumbertHumbert7]

Do you have the documentation, I’m not familiar with the Nitche method

[colinjcotter]

It isn’t documented, because Firedrake documentation is about how to express your chosen FE discretisation, not which discretisation to use for which problem. For that you will need to do a literature search.

From: HumbertHumbert7 @.> Date: Friday, 15 March 2024 at 17:03 To: firedrakeproject/firedrake @.> Cc: Cotter, Colin J @.>, Comment @.> Subject: Re: [firedrakeproject/firedrake] 4-th order ODE (Issue #3456)

Do you have the documentation, I’m not familiar with the Nitche method

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[HumbertHumbert7]

Ok, thank you but there is not an example inthe firedrake documentation with it?

[colinjcotter]

There is a formula in this paper

Kirby, Robert C., and Lawrence Mitchell. "Code generation for generally mapped finite elements." ACM Transactions on Mathematical Software (TOMS) 45, no. 4 (2019): 1-23.

On 15 Mar 2024, at 17:10, HumbertHumbert7 @.***> wrote:



If I have understood correctly, I just add a term that in this case is ufd.grad(v)[0]fd.ds?

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[HumbertHumbert7]

I check it now, thank you very much

[Ig-dolci]

@HumbertHumbert7 can we close this issue?