There are many resources available on the internet. My following queries might be succinctly explained for people with math major, but may not be for people with engineering background. It would be great if someone could clarify as I read the book to have a solid understanding.
1.) In Finite Element Method (FEM), what is meant by function space? Does function space require both physical domain, type and order of polynomial? An example would be great to illustrate the concept.
Like in page 3, of the fenics book, it is mentioned that
“Suitable function spaces must be specified for the test and trial functions.”
2.) What is meant by Sobolev space? p.3
“The solution of the underlying PDE must lie in a function space where also the derivatives are continuous, but the Sobolev space $H^{1}(\Omega)$ allows functions with discontinuous derivatives.” Could someone provide a concrete example? Does it mean that I can find solution of a function with discontinuous values at the boundary?
For example, we know the electric field inside metal is zero, but outside, let’s say in air can have a finite value. We can find the force, which is (some factors) multiplied by the gradient of the square of the electric field can be computed? I am trying to connect the dot what is mentioned by the sentence in the book.
E = -\nabla V f = \frac{1}{2} \epsilon \frac{\partial^2 E}{\partial x^2}
3.) In Eq(1.7), the trial space V is defined as V = {v \in H^{1}(\Omega) : v = u_{0} \text{ on } \partial \Omega }
Will v be u here meaning to be clearly spell out V = {u \in H^{1}(\Omega) : u = u_{0} \text{ on } \partial \Omega } ? It is most likely that I have mixed up or did not properly understood the defintion and space of trial and test functions.
As there are many resources available online, I would like to refer you to a selection of sources you can study.
A function space in mathematics is: a set of functions between two fixed sets. As it is a set, we chose the a function space with a given set of properties (such that integrable first order weak derivatives). As you can see by the definition in the FEniCS setting, we require a discrete domain (mesh), a given type of functions (such as “CG” or “DG”) and what order these functions should be of. A selection of finite elements can be found at: http://femtable.org/
Below I have listed some sources that might be of interest for you, a very short intro to the finite element method can be found here: COMSOL intro to FEM
If you want more in-depth understanding of the finite element method, I would suggest sources such as:
which is a part of a course for the FEM given at the University of Oslo (rest of course material is available here).
or Notes of the finite element method part of another course at UiO. These courses are aimed at graduate students in applied mathematics.
@dokken: Thanks for sharing useful information. The two (lecture) notes and the website that you shared are in my next pipeline for reading and understanding.
1.) So far what I read from the materials you have shared, could I say that this function
f(x) = a, \quad x>0
f(x) = 0, \quad x<0
is not continuous at x=0 and also not diffentiable at x=0 in the classical sense. However, as per my reading in wikipedia link https://en.wikipedia.org/wiki/Sobolev_space, the weak derivative of f(x) at x = 0 is g^{'}(x) = \frac{\partial (f^2)}{\partial x} = a^2, \quad \text{at } x = 0.
2.) Is there a small canonical example you could refer or provide for me to code and see the difference that the trial and test function spaces could lie in different function spaces The trial function u lies in some (possibly different) function space $\hat{V}$. We refer to (1.5) as the weak form of the original boundary-value problem (1.1). in page 3. For your reference, the weak form of the Poisson Eq.(1.5) is
\int_{\Omega} \nabla u \cdot \nabla v \,d x = \int_{\Omega} f v\,d x
@bhaveshshrimali: Thanks for providing that. But please note that I am not sure, if I am up to that level to digest the notes by Arnold. You gotta start low for me
Most often you chose the test and trial function from the same FunctionSpace. However, one usually enforce different trace-properties on the test and trial function (on the mathematical level).
The trial function will often satisfy the Dirichlet-condition u=g on the dirichlet boundary, while the test function will be 0 at this boundary. Choosing them in this way will remove integrals over the Dirichlet boundary when doing integration by parts.
@nate: Thanks. I also think that many of my queries will be addressed in this book. After going throught the fenics book, I will read especially chapter one.
