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.