Galerkin method in abstract settings

 
This section deals with analyzing errors in finite element method. To determine when Galerkin method will produce a good approximation, abstraction is introduced.
Let $B:V \times V \rightarrow R$ be bounded bilinear form and let $F:V\rightarrow R$ be bounded linear form.
It is assumed that the problem to be solved can be stated as, find $u \in V$ such that
\begin{equation}
  B(u,v)=F(v), v \in V
\end{equation}
  Example
To make sense of above abstraction, it is best to see how this is derived using a weak form of PDE as shown below.
Consider Poisson's equation given by:
\[
- \Delta u = f \quad \text{in } \Omega, \quad u = 0 \text{on } \partial \Omega
\]
where \(\Omega\) is a bounded domain, \(f\) is a given function, and \(u\) is the function to be determined.


The weak formulation involves multiplying the differential equation by a test function \( v \) from the space \( H_0^1(\Omega) \), integrating over the domain \(\Omega\), and applying integration by parts:
\[
\int_\Omega \nabla u \cdot \nabla v \, dx = \int_\Omega f v \, dx
\]
Here, \( B(u, v) = \int_\Omega \nabla u \cdot \nabla v \, dx \) and \( F(v) = \int_\Omega f v \, dx \) define the bounded bilinear and linear forms respectively.

Well posed problem
The abstract setting problem is considered well-posed, if for each $F \in V^*$ , there exists a unique solution $u \in V$ and the mapping $F->u$ is bounded. $V^*$ here means dual space of $V$. Alternately, if $L:V \rightarrow V^*$ given by $<Lu,v>=B(u,v)$ is an isomorphism.

Example
For Dirichelet problem of Poisson's equation, we have
\begin{align*}
  V=\circ{H^1}(\Sigma) \\
  B(u,v) = \int_{\Sigma} (grad u(x))(grad v(x)) dx \\
  F(v) = \int_{\Sigma} f(x) v(x) dx
\end{align*}
 A generalized Garlekin method for the abstract problem begins with a finite dimensional normed vector space $V_h$, a bileaner form $B_h:V_h \times V_h \rightarrow \mathcal{R}$ and a linear form $F_h:V_h \rightarrow \mathcal{R}$ and defines $u_h \in V_h$ by
 \begin{equation}
   B_h(u_h,v) = F_h(v), v \in V_h
 \end{equation}
 Above equation can be written in the form $L_h u_h = F_h$ where $L_h:V_h\rightarrow V_h^*$ given by $<L_hu,v>=B_h(u,v)$
 If the finite dimensional problem is non-singular, then the norm of discrete solution operator known as ``stability constant'' is defined as
 \begin{equation}
   \parallel L_h^{-1} \parallel
 \end{equation}

 In this approximation of the original problem determined by $V,B,F$ by $V_h,B_h,F_h$ intention is that $V_h$ in some sense approximates $V$ and $B_h,F_h$ approximate $B,V$. This is idea behind ``Consistency''.
 The goal here is to approximate $u_h$ to $u$. This is known as ``Convergence''.

 To this end, assume there is a restriction operator $\pi_h:V \rightarrow V_h$ so that $\pi_hu$ is close to $u$.
 Using the equation $L_h u_h = F_h$, we can compute the ``consistency error'' as
 \begin{equation}
   L_h\pi u - F_h
 \end{equation}
 Error we wish to control is
 \begin{equation}
   \pi_h u - u
 \end{equation}
 Easy to see the relation between error and consistency error.
 \begin{equation}
   \pi_h u - u = L_h^{-1}(L_h\pi u - F_h)
 \end{equation}
 Recall thet norm of $L_h$ is stability constant. If we take norms both sides to above equation, we see that the norm of error is bounded by product of stability constant and norm of consistency error.
 \begin{equation}
   \parallel  \pi_h u - u  \parallel \leq \parallel L_h^{-1} \parallel \parallel (L_h\pi u - F_h) \parallel
 \end{equation}
Expressing this in terms of bilinear forms the relation becomes,
\begin{equation}
  \parallel L_h\pi_hu - V_h \parallel = sup_{0 \neq v \in V_h}\frac{B_h(\pi_hu,v) - F_h(v))}{\parallel v \parallel}
\end{equation}
Above equation, especially RHS needs some explanation.
The consistency error measures how close $B_h(\pi_hu,v)$ to $F_h(v)$ for all test functions $v$ in the subspace $V_h$. Here $F_h$ represents the discrete analog of forcing function. The ratio including sup means worst or max consistency error to the norm of test function $v$ over all possible non-zero test functions in $V_h$.
Finite dimensional problem is non-singular iff
\begin{equation}
  \gamma_h = inf_{0\neq u \in V_h} sup_{0 \neq v \in V_h} \frac{B_h(u,v)}{\parallel u \parallel \parallel v \parallel}
\end{equation}
And the stability constant is given by $\gamma^{-1}_h$
Notes:

Above $\gamma_h$ measures smallest ration of bilinear form $B_h(u,v)$ to the product of norms of $u$ and $v$ overall non-zero functions $u,v$ in the subspace $V_h$. Above condition shows that the bilinear form $B_h(u,v)$ is bounded from below and has a positive lower bound. This is also called ``coercive'' over subspace $V_h$. Since $B_h(u,v)$ is a matrix, above condition shows that this matrix is non-singular. For numerical methods, this means the problem is well-posed and having a continuous solution.