Numerical Solution of the Nonlinear Boundary Value Problem

Abstract

The finite element method is based on the weak formulation (2.5). We consider $N\in \mathbb {N},$ $N\ge 2,$ nodes $\left\{ z_j\right\} _{j=1}^N$ such that $-2\pi \delta =:z_1<z_2<\ldots<z_{N-1}<z_N:=2\pi \delta ,$ and define the subintervals $\mathscr {I}_j:=\left( z_j,z_{j+1}\right)$ with the lengths $h_j:=z_{j+1}-z_j$ and the parameter $h:=\max _{j\in \{1,\ldots ,N-1\}}h_j.$ Then, for $j\in \{1,\ldots ,N\}$ we introduce the basis functions $\psi _j:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {R}$ by the formula

$$ \psi _j(z) :=\left\{ \begin{matrix} \left( z-z_{j-1}\right) /{h_{{\;j-1}}},&{} z\in \mathscr {I}_{j-1} \text{ and } j\ge 2,\\ \left( z_{j+1}-z\right) /{h_j},&{} z\in \mathscr {I}_j \text{ and } j\le N-1,\\ 0,&{} \text{ otherwise } \end{matrix}\right. $$

and the corresponding linear spaces

$$ V_h:=\mathop {\mathrm {span}}\{\psi _j\}_{j=1}^N :=\Big \{v_h=\sum _{j=1}^N\lambda _j\psi _j:\; \lambda _j\in \mathbb {C}\Big \}, \qquad {\mathbf V }_h:=V_h^3. $$