The convergence of the finite element method
The following three lemmas apply as follows. The Lax-Milgram lemma and the more general infimum-supremum condition are used to check whether a given weak formulation has a unique solution. The Cea lemma allows for the estimation of the solution error and the rate of convergence for the finite element method.
Let \( V=\{ v \in L^2(\Omega):\int_{\Omega} \| v \|^2 +\| \nabla v \|^2 dx_1 dx_2 < \infty, tr(v)=0 \textrm{ on } \Gamma_D \} \) for \( \Omega \subset {\cal R }^d \)
Let \( B:V\times V \rightarrow {\cal R} \)
will be a two-line form \( B(\alpha_1 u_1+\alpha_2 u_2,v_1 )=\alpha_1 B(u_1,v )+\alpha_2 B(u_2,v_1 ) \)
\( B(u_1,\beta_1 v_1+\beta_2 v_2 ) = \beta_1 B(u_1,v_1 )+ \beta_2 B(u_1,v_2 ) \quad \alpha_1,\alpha_2,\beta_1,\beta_2 \in {\cal R}, \forall u_1,u_2,v_1,v_2 \in V \),
ciągłą \( \exists M > 0, |B(u,v)| \leq M \|u\|_V \|v\|_V \forall u,v \in V \)
and coercive, i.e. \( \exists \alpha > 0, |B(u,u)| \geq \alpha \|u\|^2_V \forall u \in V \).
Let \( L:V \rightarrow {\cal R} \) will be a linear form
\( L(\alpha_1 v_1+\alpha_2 v_2 )=\alpha_1 L(v_1 )+\alpha_2 L(v_2 )\quad \alpha_1,\alpha_2\in {\cal R}, \forall v_1,v_2 \in V \), continuous, i.e.
\( \exists C > 0, |L(v)| \leq C \|v\|_V \forall v \in V \).
Then the weak problem \( B(u,v)=L(v)-B(\hat{u},v) \quad \forall v \in V \)
has a unique solution, such that \( \|u+\hat{u} \|_V \leq \|\hat{u}\|_V \left(1+\frac{M}{\alpha} \right)+\frac{C}{\alpha} \).
For more complicated cases of weak formulations \( B(u,v)=L(v)-B(\hat{u},v) \quad \forall v \in V \) instead of the Lax-Milgram Lemma, we use the Babuska Lemma equivalent to the Brezzi Lemma [1].
Consider the solution to the problem of the finite element method on a computational mesh.
Let \( V=H^1_0(\Omega) = \{ v \in L^2(\Omega):\int_{\Omega} \| v \|^2 +\| \nabla v \|^2 dx_1 dx_2 < \infty, tr(v)=0 \textrm{ on } \Gamma_D \} \) for \( \Omega \subset {\cal R }^d \)
Let \( B:V\times V \rightarrow {\cal R} \)
be a two-line form \( B(\alpha_1 u_1+\alpha_2 u_2,v_1 )=\alpha_1 B(u_1,v )+\alpha_2 B(u_2,v_1 ) \)
\( B(u_1,\beta_1 v_1+\beta_2 v_2 ) = \beta_1 B(u_1,v_1 )+ \beta_2 B(u_1,v_2 ) \quad \alpha_1,\alpha_2,\beta_1,\beta_2 \in {\cal R}, \forall u_1,u_2,v_1,v_2 \in V \),
continuous \( \exists M > 0, |B(u,v)| \leq M \|u\|_V \|v\|_V \forall u,v \in V \)
and coercive \( \exists \alpha > 0, |B(u,u)| \geq \alpha \|u\|^2_V \forall u \in V \).
Let \( L:V \rightarrow {\cal R} \) be a line form.
\( L(\alpha_1 v_1+\alpha_2 v_2 )=\alpha_1 L(v_1 )+\alpha_2 L(v_2 ) \quad \alpha_1,\alpha_2\in {\cal R}, \forall v_1,v_2 \in V \),
continuous \( \exists C > 0, |L(v)| \leq C \|v\|_V \forall v \in V \).
Let \( u \in V \) be a solution of a weak problem \( B(u,v)=L(v)-B(\hat{u},v) \quad \forall v \in V \).
Let \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) be a computational grid.
Let \( V_{hp} \subset V \) be an approximation space on a computational grid.
Let \( u_{hp} \in V_{hp} \) be a solution to the problem of the finite element method on a computational mesh.
\( B(u_{hp},v_{hp})=L(v_{hp})-B(\hat{u},v_{hp}) \quad \forall v_{hp} \in V_{hp} \) on the computational grid.
Then
\( \|u-u_{hp}\|_V \leq \frac{M}{\alpha} min_{w_{hp} \in V_{hp} } \| u-w_{hp} \|_V \).
The meaning of the Cea Lemma is as follows. Ideally, the distance of the solution of the finite element method problem \( u_{hp} \in V_{hp} \) on the computational grid \( T_{hp }=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) from the solution of the weak problem \( u \in V \) would be the minimum distance of solution \( u \in V \) of a weak problem from all elements of space \( V_h \) in which we are looking for a solution to the problem of the finite element method. It would be if constant \( \frac{M}{\alpha} =1 \). However it is \( \frac{M}{\alpha} \geq 1 \). This means that the error we make when solving the problem with the finite element method is burdened with an error resulting from the construction of the computational mesh. \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) (adaptation of the computational mesh increases the size of the space \( V_{hp} \) and makes our approximate solution \( u_{hp} \in V_{hp} \) closer to the perfect solution \( u \in V \). The remaining error is due to the ratio of the constants \( M \) continuity and \( {\alpha} \) coercivity of the bilinear functional \( B:V\times V \rightarrow {\cal R } \). This error can be eliminated by using the stabilization methods described in the relevant modules of the manual.