For parabolic equations in one space variable with a strongly coercive self-adjoint $2m$th order spatial operator, a $k$th degree Faedo-Galerkin method is developed which has local convergence of order $2(k + 1 - m)$ at the knots for the first $m - 1$ spatial derivatives and, if $k \geqq 2m$, convergence of order $k + 2$ at specific interior nodal points. These nodal points are the zeros of the Jacobi polynomial $P^{m, m}_n(\sigma) (n = k + 1 - 2m)$ shifted to the segments of the partition. All these convergence properties are preserved if suitable quadrature rules are used.

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SIAM Journal on Numerical Analysis

Bakker, M. (1982). Galerkin. methods for even-order parabolic. equations in one space variable. SIAM Journal on Numerical Analysis, 19(3), 571–587.