  We put forward a new type of spectral method for the direct numerical
simulation of flows where anisotropy or very fine boundary layers are present.
The mean idea is to take advantage of the fact that such structures are
dissipative and that their presence should reduce the number of degrees of
freedom of the flow, when paradoxically, their fine resolution incurs extra
computational cost in most current methods. The principle of this method is to
use a functional basis with elements that already include these fine structure
so as to avoid these extra costs. This leads us to develop an algorithm to
implement a spectral method for arbitrary functional bases, and in particular,
non-orthogonal ones. We construct a basic implementation of this algorithm to
simulate Magnetohydrodynamic (MHD) channel flows with an externally imposed,
transverse magnetic field, where very thin boundary layers are known to develop
along the channel walls. In this case, the sought functional basis can be built
out of the eigenfunctions of the dissipation operator, which incorporate these
boundary layers, and it turns out to be non-orthogonal. We validate this new
scheme against numerical simulations of freely decaying MHD turbulence based on
a Finite Volume code and it is found to provide accurate results. Its ability
to fully resolve wall-bounded turbulence with a number of modes close to that
required by the dynamics is demonstrated on a simple example. This opens the
way to full blown simulations of MHD turbulence under very high magnetic
fields, which were until now too computationally expensive, as the
computational cost of the proposed method, in contrast to traditional methods,
does not depend on the intensity of the magnetic field.
