FEM: How the weak form is related to an inner product

In summary, the conversation discusses a math student's dissertation on the finite element method, specifically focusing on proving that a(u,v) and l(v) are coercive and continuous. The student is struggling to understand how to relate these forms to the L2 norm, but Poincare's inequality and embedding theorems may be useful in solving the problem.
  • #1
Carla White
1
0
Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though.

I have the weak form as a(u,v)=l(v) where:
$$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$
and
$$l(v)=\int_\Omega fv-\int_{\partial\Omega_N} g_Nv$$.
I am trying to prove that a(u,v) is coercive and continuous and l(v) is continuous. I know to do this I need to relate these to an inner product with norm, I'm pretty sure this should be the L2 norm:
$$||u||_2:=\left(\int_\Omega u^2\right)^{1/2}$$.
I'm struggling to understand how these relate though, and how I get from the linear and bilinear forms to the normed form. Could somebody either explain how it works or point me in the direction of somewhere that explains it.

Thanks
Carla
 
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  • #2
Hi Carla,
I am actually working on a similar problem right now.
Poincare's inequality is a good place to start.
I think that embedding theorems are useful for the second part as well. I haven't worked it all the way through yet, but I think if you can show that the gradient is in a closed and bounded space, e.g. ##H^1_0(\Omega)## and that space is also contained within ##L^2(\Omega)## you can conclude that the variational formulation is coercive.
 

1. What is the weak form in FEM?

The weak form in FEM (Finite Element Method) is a mathematical representation of a partial differential equation (PDE) that is used to approximate a solution to the PDE. It is called "weak" because the solution is only required to satisfy the PDE in a weak sense, meaning it does not have to be differentiable.

2. What is an inner product in FEM?

In FEM, an inner product is a mathematical operation that takes two functions as inputs and produces a scalar value as output. It is used to define the norm and inner product space in which the weak form is formulated.

3. How is the weak form related to an inner product in FEM?

The weak form in FEM is derived using an inner product. This is because the inner product allows us to express the PDE in a variational form, which is necessary for the weak form. The weak form is essentially an equivalent representation of the PDE in terms of the inner product.

4. Why is the weak form used in FEM?

The weak form is used in FEM because it allows for a more general and flexible approach to solving PDEs. It also makes it possible to use piecewise continuous functions to approximate the solution, which is computationally more efficient than using differentiable functions.

5. What are the advantages of using the weak form in FEM?

Some advantages of using the weak form in FEM include: a wider range of possible approximations, better accuracy for certain types of PDEs, easier implementation of boundary conditions, and better handling of discontinuities in the solution.

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