Study guide: Introduction to finite element methods

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The linear system associated with the abstract form

$a(u,v) = L(v)\quad \forall v\in V\quad\Leftrightarrow\quad a(u,\baspsi_i) = L(\baspsi_i)\quad i\in\If$

We can now derive the corresponding linear system once and for all by inserting $u = B + \sum_jc_j\baspsi_j$ : $a(B + \sum_{j\in\If} c_j \baspsi_j,\baspsi_i)c_j = L(\baspsi_i)\quad i\in\If$

Because of linearity,

$\sum_{j\in\If} \underbrace{a(\baspsi_j,\baspsi_i)}_{A_{i,j}}c_j = \underbrace{L(\baspsi_i) - a(B,\baspsi_i)}_{b_i}\quad i\in\If$