9.1 The Fundamentals Behind MultiRegions
Up to now in the library, the various data structures and methods associated with
standard regions, spatial domains and local regions are not specifically dictated by any
particular numerical method. In fact, at this stage, they can all be viewed in light
of approximation theory. With local regions in place, we have a region in world
space over which we can represent an expansion (i.e. linear combination of basis
functions) and form its derivatives and its integral. It is at the level of MultiRegions that
we now combine two fundamental concepts: the idea of a collection of elements
together to form a “global" expansion and the idea of how these (local) elements
communicate (in the sense of how does one form approximations of a PDE of these
collections of elements). Hence MultiRegions is important because it gives us a way
of dealing with general tessellation and also because it is the first place at which
we can connect to a specific numerical PDE approximation methods of choice (i.e.
continuous Galerkin FEM methods, discontinuous Galerkin finite volume methods,
Because MultiRegions is both about grouping of elements in space (to form a domain) and
about solving PDEs over these domains, you will find two primary collections of routines
contained within the MultiRegions directory. You will find things related to collections of
elements: ExpList (Expansion List), DisContField (discontinuous field) and ConField
(continuous field); and you will find objects related to the linear systems formed based upon
the particular numerical method once selects (i.e. GlobalLinSys, which stands for Global
At present, the Nektar++ framework supports three types of numerical PDE discretizations
for conservation laws:
- Discontinuous Galerkin Methods: These weak-form (variational) methods do not
require element continuity, but do put restrictions on the flux of information
between elements. In general, these methods can be thought of as being in the
class of finite volume (FV) methods. One feature of these methods that is often
exploited computationally is that many operations can be considered as elemental.
See [20, 39] and references therein for a more complete summary.
- Continuous Galkerin Methods: These weak-form (variational) methods require at
least C0 continuity. Mathematically, there have been extensions to higher levels of
continuity, e.g. Isogeometric Analysis , these are not implemented in Nektar++
and would require further constraints on our SpatialDomain representations than
we currently accommodate. In general, these methods can be thought of as being in
the class of finite element methods (FEM). Although these methods are technically
(mathematically) formulated as global methods, their elemental construction and
compact basis types allow for many local operations. Many (but not all) of the
linear system routines that are contained with the MultiRegions directory are
focussed on this discretization type. See [57, 25, 44] and references therein for a
more complete summary.
- Flux Reconstruction Methods: These strong-form methods do not require element
continuity, but like dG methods they impose restrictions on the flux of information
between elements. In general, these methods can be though of as being in the
class of generalized finite difference (FD) or collocating methods. See [47, 66] and
references therein for a more complete summary.
Figure 9.1 Diagram to help explain assembly. UPDATE.
Figure 9.2 Diagram to help explain assembly. UPDATE.