5.1. Finite Element Methods for Two-Dimensional Problems
109
T ABLE 5.2. Number of basis functions associated with
each node.
Node
Undetermined coefficients
Basis functions
1
Cl. Cs• Cs
tPh tPs. tPs
2
C2• C6• C9
tP2. tP6' tP9
3
C3• C7• ClO
tP3' tP7. tPlO
4
C4
tP4
As we now have the expressions of basis functions for each element, the
coefficients ofEq. (5.1.12) can easily be calculated. Three solutions associated
with each vertex are obtained.
These solutions are the value of the unknown function, and the values of
its partial derivatives with respect to x and y at the node. The solution
associated with the center node is just the value of the unknown function at
that point.
The solution obtained by the Hermite element method has continuous
first-order partial derivatives at the vertices. This is the main reason of
choosing the Hermite element method.
When the flow equation is solved by the FEM with linear elements, the
flow velocity obtained by Darcy's law is discontinuous along the sides of
elements, because the gradient field of the numerical solution is piecewise
constant. In order to avoid this problem, some authors proposed using the
third-order Hermite elements to solve the flow equation, so that the continuity of flow velo city can be maintained, and the accuracy of the numerical
solution of the advection-dispersion equation may be improved.
Recently, Cordes and Kinzelbach (1992) presented an approach for generating highIy-accurate velocity fields for mass transport calculation when
linear, quadratic and cubic FEM are used.
5.1.4 Isoparametrie Finite Elements
Figure 5.9 shows a quadrilateral element with curved sides and seven nodes.
The sides connected with nodes 1 and 2, as weH as 6 and 7, are linear; the side
with nodes 1, 4, and 7 is quadratic; the side with nodes 2, 3, 5, and 6 is of the
third order. It is obvious that elements of such a shape are very suitable for
representing the boundaries of an irregularly shaped domain or the inner
borders in a non-homogeneous medium. We now wish to transform the
element into a quadrilateral in the e11 plane of a local coordinate system
through the transformation x = x(e, 11), y = y(e, 11)· The positions of related
nodes are shown in Figure 5.9.
This transformation can be constructed using basis functions. When node i
is located on the side of the element, as nodes 3, 4, 5 in Figure 5.9, its basis
109
T ABLE 5.2. Number of basis functions associated with
each node.
Node
Undetermined coefficients
Basis functions
1
Cl. Cs• Cs
tPh tPs. tPs
2
C2• C6• C9
tP2. tP6' tP9
3
C3• C7• ClO
tP3' tP7. tPlO
4
C4
tP4
As we now have the expressions of basis functions for each element, the
coefficients ofEq. (5.1.12) can easily be calculated. Three solutions associated
with each vertex are obtained.
These solutions are the value of the unknown function, and the values of
its partial derivatives with respect to x and y at the node. The solution
associated with the center node is just the value of the unknown function at
that point.
The solution obtained by the Hermite element method has continuous
first-order partial derivatives at the vertices. This is the main reason of
choosing the Hermite element method.
When the flow equation is solved by the FEM with linear elements, the
flow velocity obtained by Darcy's law is discontinuous along the sides of
elements, because the gradient field of the numerical solution is piecewise
constant. In order to avoid this problem, some authors proposed using the
third-order Hermite elements to solve the flow equation, so that the continuity of flow velo city can be maintained, and the accuracy of the numerical
solution of the advection-dispersion equation may be improved.
Recently, Cordes and Kinzelbach (1992) presented an approach for generating highIy-accurate velocity fields for mass transport calculation when
linear, quadratic and cubic FEM are used.
5.1.4 Isoparametrie Finite Elements
Figure 5.9 shows a quadrilateral element with curved sides and seven nodes.
The sides connected with nodes 1 and 2, as weH as 6 and 7, are linear; the side
with nodes 1, 4, and 7 is quadratic; the side with nodes 2, 3, 5, and 6 is of the
third order. It is obvious that elements of such a shape are very suitable for
representing the boundaries of an irregularly shaped domain or the inner
borders in a non-homogeneous medium. We now wish to transform the
element into a quadrilateral in the e11 plane of a local coordinate system
through the transformation x = x(e, 11), y = y(e, 11)· The positions of related
nodes are shown in Figure 5.9.
This transformation can be constructed using basis functions. When node i
is located on the side of the element, as nodes 3, 4, 5 in Figure 5.9, its basis