Яревский Е.А. Численные методы для дифференциальных уравнений в частных производных

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values at two corresponding vertexes only. Therefore, the basis function are

continuous.

The biquadratic Lagrange elemental functions Any function U(ξ, η)

on the rectangle is approximated with the biquadratic basis as

U(ξ, η) =

i=1

j=1

i,j

(ξ, η),

where

i,j

(ξ, η) = N

(ξ)N

(η), i, j = 1, 2, 3.

Figure 18: The biquadratic elemental functions for the vertex, edge and

surface.

The one-dimensional elemental functions are given by

(ξ) = −ξ(1 − ξ)/2, N

(ξ) = ξ(1 + ξ)/2,

(ξ) = (1 − ξ

), −1 ≤ ξ ≤ 1.

So we can see that the biquadratic function N

i,j

(ξ, η) is generally written as

i,j

(ξ, η) = a

+ a

ξ + a

η + a

+ a

ξη + a

η + a

ξη

+ a

It contains s ome terms of the order higher than two, they ale collected in the

second line. However, an arbitrary polynomial of the second degree only can

be represented by N

i,j

(ξ, η). According to Theorem 4-2, we have here the

approximation of the second order. The higher degree terms do not improve

the approximation. In fact, they are redundant, and might even result in the

degradation of the numerical accuracy and stability.

We have constructed the basis function on the computational plane, and

need to transform them into the physical plane. The mapping of the canonical

square onto the physical rectangle (x

, y

) is given with the functions N





i=1

j=1





(ξ)N

(η).

Figure 19: The mapping of the canonical square onto the physical rectangle.

For example, the edge η = −1 maps onto the edge (x

, y

) − (x

, y









1 − ξ





1 + ξ

. (54)

It is important that the vertexes (x

, y

) and (x

, y

) do not enter Eq.(54)

and do not aﬀect this transformation. Therefore, the continuity of the basis

functions is preserved with this mapping.

Lecture 6

Multidimensional FEM. The hierarchical ﬁnite elements. Three-dimensional

elements. The interpolation errors.

16 The hierarchical elements

In order to study the methods for the construction of the multidimensional

hierarchical ﬁnite elements, we restrict ourselves to the simpler case of the

rectangles. We describe the hierarchical basis for the square. These functions

can be then recalculated for an arbitrary rectangle with the transformation

described in the previous lecture.

The order p basis consists of the diﬀerent types of functions. Those

functions are linked with the vertices, edges and the center of the square.

Let us describe them.

Elemental functions for the vertices (4 bilinear functions):

(ξ, η) = N

(ξ)N

(η), i, j = 1, 2.

Elemental functions for the edges (linked with the center of edges, 4(p − 1)

functions existing for p ≥ 2):

(ξ, η) = N

(η)N

(ξ), N

(ξ, η) = N

(ξ)N

(η),

(ξ, η) = N

(η)N

(ξ), N

(ξ, η) = N

(ξ)N

(η).

In the last equations the index k spans the values k = 2, 3...p. The function

(y) are deﬁned as

(y) =

2k − 1

−1

k−1

(t) dt.

Here P

k−1

(t) is the Legendre polynomial of the de gree k − 1. It is easy to

check that N

(−1) = N

(1) = 0, so each elemental function for the edges is

identically zero on three edges out of four. Few of these functions are plotted

in Fig. 20.

Elemental functions linked with the center of the square (3,3). These (p −

2)(p − 3)/2 functions existing for p ≥ 4) can be described in terms of the

function N

400

= (1 − ξ

)(1 − η

For p = 4, only this f unction enters the expansion. For the polynomial

degrees p = 5, 6, the hierarchical functions are deﬁned as

510

= N

400

(ξ), N

501

= N

400

(η),

620

= N

400

(ξ), N

611

= N

400

(ξ)P

(η), N

602

= N

400

(η).

The upper index kλµ of the function consists of the total polynomial degree

and the degrees of the additional polynomials in ξ and η coordinates so that

k = λ + µ + 4. Few of these functions are plotted in Fig. 21. For the higher

polynomial degrees p > 6, the functions are introduced in the similar way.

Figure 20: The hierarchical functions for the vertex and for the edges (k =

1, 2, 3, 4).

The solution in terms of the hierarchical basis is expanded as follows:

U(ξ, η) =

i=1

j=1

k=2

j=1

i=1

k=4

λ+µ=k−4

kλµ

The total number of the functions in this representation can be calculated as

4 + 4(p − 1)

(p − 2)

(p − 3)

, where q

= max(q, 0).

It is interesting to compare the total number of the function for three

basises of the order p: the direct product basis and the hierarchical basis in

the canonical square, and the minimally admissible set which coincides with

the basis in the canonical triangle. These number are presented in the table:

degree p 1 2 3 4

the triangle basis 3 6 10 15

the direct product basis 4 9 16 25

the hierarchical basis 4 8 12 17

One can see that both basises for the square are not optimal with respect to

the number of functions. For the low polynomial degrees p = 1, 2, they are

very similar, and the number of the functions essentially exceeds that for the

triangle. For higher p, however, quality of the hierarchical basis improves

and the number of the functions asymptotically converges to the optimal

value. Quality of the direct product basis stays the same, the number of the

functions essentially exceeds the optimal value.

17 The three-dimensional elements

17.1 The tetrahedron

For the three-dimensional case, the number of polynomials of the degree p is

equal to

(p + 1)(p + 2)(p + 3)

Figure 21: The hierarchical elemental functions linked with the center of the

square N

kλµ

, λ + µ = k − 4 (k = 4, 5, 6).

The construction of the elemental functions is done in the same way as for

the two-dimensional case s o we give here the formulas for the linear Lagrange

elements only.

For the tetrahedron, the Lagrange conditions

, y

, z

) = δ

, j, k = 1, 2, 3, 4,

lead to the following representation for the linear functions:

(x, y, z) =

klm

(x, y, z)

jklm

Here (jklm) is a permutation of (1,2,3,4) numbers, and the determinants

klm

and C

jklm

are written as

klm

= det







1 x y z

1 x







, C

jklm

= det







1 x







Figure 22: The nodes for the linear elemental function on the tetrahedron

and barycentric coordinates.

The projection of the solution U on the element can be expanded in terms

of these functions as

U(x, y, z) =

j=1

(x, y, z).

The barycentric coordinates (also known as the volume coordinates) are

deﬁned as

= N

(x, y, z), j = 1, 2, 3, 4.

These coordinates also give the volumes of the corresponding tetrahedrons

with the vertex at the point P :

P 234

1234

, ζ

P 134

1234

, ζ

P 124

1234

, ζ

P 123

1234

As in the two-dimensional case, the barycentric coordinates are redundant.

The inverse coordinate transformation can be deﬁned with the formula



















1 1 1 1



















The last line in this equation shows us the redundancy of the barycentric

coordinates.

Figure 23: The nodes for the quadratic and cubic elemental functions on the

tetrahedron.

17.2 The cube

The canonical cube is deﬁned as

{(ξ, η, ζ) : −1 ≤ ξ, η, ζ ≤ 1}.

The elemental function for the node (ijk) is written as the direct product of

the one-dimensional linear functions:

ijk

= N

(ξ)N

(η)N

(ζ).

18 The interpolation error

The estimation of the interpolation errors will b e done in two steps:

• the interpolation error estimation with polynomials on the canonical

element

• the recalculation of the interpolation errors from the canonical to the

physical element.

Figure 24: The nodes for the tri-linear tri-quadratic elemental functions on

the cube.

For the sake of simplicity, let us consider the two-dimensional case. We

expand a function on the canonical element in terms of the elemental func-

tions

U(ξ, η) =

j=1

(ξ, η). (55)

Then the interpolation error on the canonical element is estimated with the

following

Theorem 1: Let p be the maximal integer number, such that Eq.(55) is exact

for any polynimial of the degree p. Then for the canonical element Ω

there

exists C > 0 such that

|u − U|

s,Ω

≤ C|u|

p+1,Ω

∀u ∈ H

p+1

(Ω

), s = 0, 1...p + 1.

where |u|

s,Ω

stands for the Sobolev quasinorm.