2  Finite Element Method

2.1 The strong form

Consider the 1D Poisson equation as a BVP on the domain \(\Omega\) \[ (-\kappa u_x)_x = f \] \[ x \in \Omega = (0, L) \] \[ u(0)=g_0, u(L)=g_L \]

This is the strong form of the BVP.

Strong form

The solution \(u\) satisfies the BVP in a point-wise manner, i.e. at every point in the continuous domain.

Well-posedness

The BVP is well-posed iff there exists a unique solution which depends continuously on the problem data. For the BVP to be well-posed, the following conditions must be met: \[\kappa \in C^1([0,L])\] \[f \in C^0([0,L])\]

2.2 Weak form

A residual of an equation in \(u\) is the difference

\[ R(u) = (-\kappa u_x)_x - f \]

A weighted residual is \[ \int_\Omega w R(u) d\Omega \quad \forall w \in W, \] where \(W\) is some space of weighting (test) functions.

“Test” functions and terminology

The functions \(w\) are “test functions” because they provide a test for the solution in the weak \(R(u)=0\) sense. Given \(v \in \mathbb{R}^n\), find \(v=0\). We can only find \(v\), \(w\) for any \(w\).

Iff \(R(u)=0\), then \(u\) solves the BVP. Thereby, if $R(u) in a point-wise manner, but nevertheless \(u\) satisfies \(R(u)=0\) in a weaker, integral, sense the BVP is called generalized to the weak problem. Instead of looking for \(R(u)=0\) at all points in \(\Omega\), we look at

\[ \int_\Omega wR(u) d\Omega = \int_\Omega w(-\kappa u_x)x d\Omega - \int_\Omega wf d\Omega \]

Alternatively, use integration by parts to move some derivatives to the weight:

\[ \int_\Omega wR(u) d\Omega = \int_\Omega \kappa u_x w_x d\Omega - \int_\Omega wf d\Omega + \text{boundary terms} \]

The Weak Form

A weak form (\(w\)) to the BVP consists of:

• Formulating a eneralized weighted residual
• Choice of a space of test functions \(W\), for which \(\int_\Omega wR(u) d\Omega = 0\)
• Choice of a space of trial functions \(S\), where we will look for the solution \(u\)

Note that the weak form is not unique as different choices for the test and trial function spaces!

The weak problem

Find \(u \in S\ \forall w \in W\)i, such that \(\int_\Omega wR(u)\ d\Omega = 0\)

Why is this called a “weak” problem?

• \(u_s\) solves the strong problem, then it also solves the weak problem
• If \(u_w\) solves the weak problem, then \(u_w\) may not solve the strong problem
• The weak form allows more general problem data, such as discontinuities in coefficients
• If \(u_w\) solves the weak problem, then \(u_w\) may not even have sufficient smoothness for the strong problem to be well-defined

Going forward, we will use the notation: \[ \int_\Omega wR(u) d\Omega = B(u,w) - L(w) \]

2.2.1 Restrictions on the trial and test space

What conditions do the trial space \(S\) and the test space \(W\) need to meet such that \(B(u,w)\) and \(L(w)\) are well-defined?

2.2.1.1 \(L(w)\)

\[ L(w) = \int_\Omega w f d\Omega \] In general, assume that \(\int_\Omega f^2 d\Omega < \infty\) and \(\int_\Omega w^2 d\Omega < \infty\).

Cauchy-Schwarz Inequality

\[ \left|\int_\Omega fg d\Omega \right| \leq \left(\int_\Omega f^2 d\Omega\right)^\frac{1}{2}\left(\int_\Omega f^2 d\Omega\right)^\frac{1}{2} \]

Through the Cauchy-Schwarz inequality, if \(w \in g: \int_\Omega g^2 d\Omega < \infty\), then \(L(w)\) is finite.

2.2.1.2 \(B(u,w)\)

\[ \left|B(u,v)\right| = \left|\int_\Omega\kappa u_{,x}w_{,x}\ d\Omega\right| \leq \max_{x\in\Omega}\kappa(x)\left|\int_\Omega u_{,x}w_{,x}\right| \leq \kappa_\text{max}\left(\int_\Omega(u_{,x})^2\right)^{\frac{1}{2}}\left(\int_\Omega(w_{,x})^2\right)^{\frac{1}{2}}\] The last part of the equation only has terms that we know are finite, \(\kappa\) since we specify it to be finite, and the Cauchy-Schwarz inequality then applies for the remaining terms.

A complete specification of the prblem in weak form is: \[ \begin{align} B,L,S &= \{ u: u, u_{,x} \in L^2(\Omega), u(0)=g, u(L)=g_L\}\\ W &= \{ w: w, w_{,x} \in L^2, w(0)=w(L)=0 \} \end{align} \] Where we then want to find \(u \in S\) such that \(B(u,w)=L(w) \forall w \in W\). Furthermre, this is called “strong enforcement” of the bundary conditions. There is also weak enforcement of the boundary conditions, which will be described later. It is also common to write \(S=W+g\), where \(g\) is any function such that \(g, g_{,x}\in L^2\), \(g(0)=g_0\) and \(g(L)=g_L\).

2.2.2 Equivalence between strong and weak form

Strong \(\rightarrow\) Weak

Let \(u\) solve the strong form. \[ \begin{align} R(u)=0 &\implies \int w R(u) = 0 \forall w \in W\\ &\implies B(u,w)=L(w)=0 \forall w \in W \quad\text{Through integration by parts} \end{align} \]

Weak \(\rightarrow\) Strong

Let \(u\) solve the weak form. \[ \begin{align} B(u,w) - L(w) &= 0 \forall w \in W\\ \int_\Omega R(u)w &= 0 \forall w \in W \implies R(u)=0 \quad\text{Through the Dubois-Reymond Lemma} \end{align} \]

2.3 Discrete Weak Form

Here we want to transform the continuous problem to a discrete one through discretization, such that we can use computers to solve the problem. This move from the continuous weak form to the discrete weak form is non-unique as more choices must be made.

In essence, we want to set \(B_h\), \(L_h\), \(S_h\), \(W_h\), where \(h\) denotes the discrete nature of the problem.

A choice we can make is to simply “take over” \(B_h\) and \(W_h\) from their continuous counterparts: There are a number of sensible options to take:

1. \[ \begin{align} B_h(w_h,w_h) &= B(u_h, w_h)\\ L_h(w_h)&=L(w_h) \end{align} \]

2. \[ \begin{align} B_h(u_h,w_h) = B(u_h,w_h) + \text{h-dependent terms} \end{align} \]

For the second option, the additional h-dependent terms can be useful to help the stability of the discrete problem. Another reason for additional terms can be to simplify imposition of boundary conditions.

The second component to the discrete weak problem is the choice of the solution and test spaces \(S_h\) and \(W_h\). The focus in these notes will be on “conforming” discretizations. This means that \(S_h \in S\) (\(W_h\) can still be chosen “freely”). The most popular choice is the Bubnov-Galerkin method, where \(W_h=S_h\) (up to boundary conditions). Other choices (i.e. where \(W_h\neq S_h\)) are called Petrov-Galerkin methods. The goal of the discretization is to make choices such that we have stability for any \(h\).

We also are looking for an approximation property, so we want to be able to use the choice of space to work for different problems, not just for a single special case. This also means that \(S_h \rightarrow S\) as \(h\rightarrow0\).

2.4 The Finite Element Method (FEM)

Finite

In FEM, we only work with finite-dimensional spaces such that computers are able to solve them

Element

We discretize \(\Omega\) into smaller subdomains called “elements”

In essence, we have a local function space \(E(\Omega_e)\) for an element \(e\). We then set \(S_h =\{f: f\|_{\Omega_e} E(\Omega_e)\ \forall e\} \cap S\).

Choosing polynomials as the basis is a very popular choice. \(F(p,k,\tau) = \{f: f\|_{\Omega_i} \in P_p\ \text{with} i=1,2,...m\). We may also set continuity requirements on the derivatives (- and + denote the direction from which we look at the derivative): \[ \frac{d^kf(x_i^-)}{dx^k}=\frac{d^kf(x_i^+)}{dx^k} \]