NotationsΒΆ

We first recall the glt symbols for the 1d mass, stiffness and advection respectively,

M^p_n &= \left[\int_0^1 N_{i_1}^p(t) ~ N_{j_1}^p(t) ~dt\right]_{i_1, j_1=1}^{n+p}, \\ S^p_n &= \left[\int_0^1 \left(N_{i_1}^p(t)\right)^{\prime} ~ \left(N_{j_1}^p(t)\right)^{\prime} ~dt\right]_{i_1, j_1=1}^{n+p}. \\ A^p_n &= \left[\int \left(N_{i_1}^p\right)^{\prime}(t) ~ N_{j_1}^p(t) ~dt\right]_{i_1, j_1=1}^{n+p},

Their corresponding GLT symbols are

\{n M^p_n\}_n & \sim_{\rm GLT} \mathfrak{m}_p \\ \{\frac{1}{n} S^p_n\}_n & \sim_{\rm GLT} \mathfrak{s}_p \\ \{ A^p_n\}_n & \sim_{\rm GLT} \mathfrak{a}_p

where,

\mathfrak{m}_p(x, \theta) &:= \mathfrak{m}_p(\theta) = \phi_{2p+1}(p+1) + 2 \sum_{k=1}^p \phi_{2p+1}(p+1-k) \cos(k \theta). \\ \mathfrak{s}_p(x, \theta) &:= \mathfrak{s}_p(\theta) = - {\phi}''_{2p+1}(p+1) - 2 \sum_{k=1}^p {\phi}''_{2p+1}(p+1-k) \cos(k \theta). \\ \mathfrak{a}_p(x, \theta) &:= \mathfrak{a}_p(\theta) = - 2 \sum_{k=1}^p {\phi}'_{2p+1}(p+1-k) \sin(k \theta).