2.6 Discretization Approaches
37
derivative of the integral with respect to each nodal value to be zero; this
corresponds to selecting the best solution within the set of allowed functions
(the one with minimum residual). The result is a set of non-linear algebraic
equations.
An important advantage of finite element methods is the ability to deal
with arbitrary geometries; there is an extensive literature devoted to the
construction of grids for finite element methods. The grids are easily refined;
each element is simply subdivided. Finite element methods are relatively easy
to analyze mathematically and can be shown to have optimality properties
for certain types of equations. The principal drawback, which is shared by any
method that uses unstructured grids, is that the matrices of the linearized
equations are not as well structured as those for regular grids making it more
difficult to find efficient solution methods. For more details on finite element
methods and their application to the Navier-Stokes equations, see books by
Oden (1972), Zinkiewicz (1977), Chung (1978), Baker (1983), Girault and
Raviart (1986) or Fletcher (1991).
A hybrid method called control-volume-based finite element method (CVF E M ) should also be mentioned. In it, shape functions are used to describe
the variation of the variables over an element. Control volumes are formed
around each node by joining the centroids of the elements. The conservation
equations in integral form are applied to these CVs in the same way as in
the finite volume method. The fluxes through CV boundaries and the source
terms are calculated element-wise. We shall give a short description of this
approach in Chap. 8.
37
derivative of the integral with respect to each nodal value to be zero; this
corresponds to selecting the best solution within the set of allowed functions
(the one with minimum residual). The result is a set of non-linear algebraic
equations.
An important advantage of finite element methods is the ability to deal
with arbitrary geometries; there is an extensive literature devoted to the
construction of grids for finite element methods. The grids are easily refined;
each element is simply subdivided. Finite element methods are relatively easy
to analyze mathematically and can be shown to have optimality properties
for certain types of equations. The principal drawback, which is shared by any
method that uses unstructured grids, is that the matrices of the linearized
equations are not as well structured as those for regular grids making it more
difficult to find efficient solution methods. For more details on finite element
methods and their application to the Navier-Stokes equations, see books by
Oden (1972), Zinkiewicz (1977), Chung (1978), Baker (1983), Girault and
Raviart (1986) or Fletcher (1991).
A hybrid method called control-volume-based finite element method (CVF E M ) should also be mentioned. In it, shape functions are used to describe
the variation of the variables over an element. Control volumes are formed
around each node by joining the centroids of the elements. The conservation
equations in integral form are applied to these CVs in the same way as in
the finite volume method. The fluxes through CV boundaries and the source
terms are calculated element-wise. We shall give a short description of this
approach in Chap. 8.