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I know that in, for example, Galerkin's method we try to approximate the solution via a sum

$$ \sum_{j=1}^{n} u_{j} a\left(e_{j}, e_{i}\right) $$

with $a\langle \cdot ,\cdot \rangle$ a bilinear form and $u_j$ are the test functions. However I have come across a sentence 'Each row of a Galerkin system matrix is associated with a locally supported test function, while each matrix column is associated with a trial function. ' and I'm not sure what the trial functions are. Can anybody help or point me to a definition?

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  • $\begingroup$ See here. They use trial functions to name the functions in the space where the solution of the integral equation are found. $\endgroup$ – user647486 Apr 20 at 15:48
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Take a (good) look at this Question & Answer @ MSE :

Then somewhere in the answer you will find the following expression: $$ \begin{bmatrix} f_1 & f_2 & f_3 & f_4 & f_5 & \cdots \end{bmatrix} \times \\ \begin{bmatrix} E_{0,0}^{(1)} & E_{0,1}^{(1)} & 0 & 0 & 0 & \cdots \\ E_{1,0}^{(1)} & E_{1,1}^{(1)}+E_{0,0}^{(2)} & E_{0,1}^{(2)} & 0 & 0 & \cdots \\ 0 & E_{1,0}^{(2)} & E_{1,1}^{(2)}+E_{0,0}^{(3)} & E_{0,1}^{(3)} & 0 & \cdots \\ 0 & 0 & E_{1,0}^{(3)} & E_{1,1}^{(3)}+E_{0,0}^{(4)} & E_{0,1}^{(4)} & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \end{bmatrix} \begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ \cdots \end{bmatrix} $$ In short: $$ \sum_i\sum_j f_i\,S_{ij}\,T_j $$ So $f_i$ is the discretization of a test function $f(x)$ and it is associated with a matrix row $(i)$;
$T_j$ is the discretization of a trial function $T(x)$ and it is associated with a matrix column $(j)$.

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