Hi,

• "Conventional DG method":

The PDE is transformed into a variational equation, e.g.

=   where denotes the interior product defined by the

integral of products of the left and right terms over the domain. This variational

equation holds for continuous spaces. It becomes discrete by restricting to

piecewise polynomials: One looks for a piecewise polynomial solution u \in U (Ansatz space) by testing against all v \in V (Test space). Using the "conventional Galerkin method" typically means that U=V, i.e., Test and Ansatz space are equal (or mutually equal, depending on the

boundary condition one must introduce small differences between U and V).

• Collocation DG method:

Instead of testing against all v \in V one checks against all Dirac-delta functions at the

the collocation points. To get a regular linear system one needs (dim U) collocations points.

Or in other words, replacing the term "Dirac-delta": the integral \int u v \dOmega

is replaced by   the point evaluation  nabla u(x_j) =  f(x_j) , for all collocation points x_j, j=1,...,n . So, there is no need for the integration over the domain. The choice of the

collacation points is a non trivial task. Usually, one take zeros of Legendre polynomials or

of Chebyshev polynomials.  Disadvantages in comparison to "conventional Galerkin":

a) Numerical analysis (uniqueness, existence, convergence of the

discrete against the continuous solution) becomes more difficult - to discuss. 

b) "Conventional": If the linear operator in the  continuous variational equation is symmetric,

one obtains a linear system Ax=b with a symmetric matrix A.

Collacation: A will not be symmetric ---> pcg-Algorithm can not be used to solve the linear system.

Andreas,

Many thanks for your comprehensive comments. Could you please introduce some references in this regard? I also have another question. To solve an elliptic equation with nonlinear terms which one is a better choice: writing the equation in weak form and use e.g. LDG method or stick with a collocation approach?  

Mohammad,

hp-Version, Galerkin (continuous but probably also the non continuous):

C. Schwab: p- and hp- Finite Element Methods

                  Theory and Applications in Solid and Fluid Mechanics

p-Version Collocation Method:

Trefethen: SPECTRAL METHODS IN MATLAB - University of Oxford

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