ABSTRACT

The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.

1. Introduction

The finite elements with inter-elemental continuous differentiability are more complicated than those providing only continuity. Such two-dimensional elements are mostly developed for triangles: the Argyris triangle [1], the Bell reduced triangle [2], the family of Morgan-Scott triangles [3], the Hsieh-Clough-Tocher macrotriangle [4], the reduced Hsieh-Clough-Tocher macrotriangle [5], the family of Douglas-Dupont-Percell-Scott triangles [6], and the Powell-Sabin macrotriangles [7]. The Fraeijs de Veubeke-Sander quadrilateral [8] and its reduced version [9] are also composed of triangles. As for single, noncomposite rectangles, the Bogner-Fox-Schmit (BFS) element [10] is the most popular and simplest one in the family of elements discribed by Zhang [11]. All these elements are widely used in the conforming finite element method for the biharmonic equation and other equations of the fourth order (see for example [12-18] and references therein) along with mixed statements of problems and a nonconforming approach [12,14,17].

A direct application of the BFS-element is restricted to the case of a simple domain that is composed of rectangles or parallelograms whose sides are parallel to two different straight lines. This condition fails even in the case of a simple polygonal domain where the intersection of the boundary with rectangles results in triangular cells (cf. Figure 1). Of course, one can construct the special triangulation compatible with the boundary as some isoparametric image [19] of a domain composed of rectangles with sides parallel to the axes. This way requires solving some additional boundary value problems for the construction of such a mapping that is smooth over the whole domain.

In this paper we suggest to use the BFS-elements in the direct way (without an isoparametric mapping) for a couple with the proposed triangular Hermite elements with 13 degrees of freedom. These triangular elements supplement BFS-elements in the following sense. They are used only near the boundary of a polygonal domain and provide inter-element continuous differentiability between finite elements of these two types. Thus, due to the joint use of these elements on a polygonal domain, the approximate solution of finite element method belongs to the class of functions which are continuously differentiable on the closure of the domain.

2. Triangulation of a Domain and the Bogner-Fox-Schmit Element

Let be a convex polygon with a boundary Assume that we can construct a triangulation of subdividing it into closed rectangular and triangular cells so that the most of them are rectangles and only a part of the cells adjacent to the boundary may be rectangular triangles. Besides, any two cells of may not have a common interior point and any two triangular cells may not have a common side. In addition, the union of all the cells coincides with A simple example of the triangulation is shown in Figure 1. We also assume that all sides of the rectangular cells and the catheti of the triangular cells are parallel to the axes.

Denote by the maximal diameter of all meshes. On rectangular cells we use the Bogner-FoxSchmit element [1]. It is defined by the triple

where is the rectangle with vertices

and

and with side lengths and

Besides, is the space of bicubic polynomials on and is the set of linear functionals (degrees of freedom or nodal parameters) of the form [12]

(1)

The dimension of (the number of the coefficients of a polynomial) is equal to 16 and coincides with the number of the degrees of freedom. For this element we have the Lagrange basis in that consists of the functions satisfying the condition

(2)

where is the Kronecker delta. These functions can be written in the explicit form with the help of the one-dimensional splines

Indeed, the direct calculations show that the basis has the form

,

3. The “Reference” Triangular Hermite Element

First we construct the “reference” triangular element with the specified properties. We consider the right triangle which has 4 nodes with the coordinates (0, 0), (1, 0), (0.5, 0.5), and (0, 1), respectively (Figure 2).

We define the space of functions and the set of degrees of freedom as follows:

(3)

Observe that at each of the nodes, and there are 4 degrees of freedom and at the node there is only one degree. Besides, the degrees of freedom for the nodes, and coincide with those for the nodes of the BFS-element (1).

Lemma 1. The triple is a finite element.

Proof. The dimension of the space coincides with the number of elements of the set Hence, to prove unisolvence of the couple, it is sufficient to construct the Lagrange basis { where for i = 1, 2, 4 and j = 1 for } on satisfying the condition [12]

(4)

The direct calculations show that the Lagrange basis has the following form:

(5)

Let be an arbitrary function. Along the side it is a polynomial of degree 3 in Together with the derivative it is uniquely determined by the values, and of nodal parameters. In addition, the derivative along is a polynomial of degree 3 in as well and is uniquely determined by the valuesand of nodal parameters. On the side

similar statements are valid within the replacement of by

Thus, on the sides and the values of a function of and its first-order partial derivatives are uniquely determined by the values of nodal parameters of at the nodes on the corresponding side.

On the side a similar property does not hold. Generally speaking, the element is not of the class. This follows from the fact that the firstorder partial derivatives of the basis functions related to the node do not vanish on the side. However, further we assume that the side of any triangular element of being the image of the hypotenuse is a part of the boundary and can not be a common side of two meshes. Hence, this feature has no influence on interelemental continuity inside the domain.

To check the interpolation properties, we use the usual notations for Sobolev spaces. Here is the Hilbert space of functions, Lebesgue measurable on equipped with the inner product

and the finite norm

For integer nonnegative is the Hilbert space of functions whose weak derivatives up to order inclusive belong to. The norm in this space is defined by the formula

We also use the seminorm

Let be an arbitrary function of By a Sobolev embedding theorem, is continuously embedded into [20], hence,. Thus, we can construct its interpolant Denote by the number of degrees of freedom related to a node We have

Theorem 1. Let Then for any integer we have the estimate

(6)

with constant independent of (and).

Proof. The maximal order of partial derivatives equals 2 in the definition of the set As mentioned above, the space is embedded into Besides, from (3) it follows that where is the space of polynomials of degree no more than 3.

Thus, all the hypotheses of the Theorem 3.1.5 in [12] are fulfilled, which implies the estimate (6). ,

4. Combination of Rectangular and Triangular Elements

Let be an arbitrary triangular element (Figure 3) with vertices

and side lengths

The affine mapping that maps the «reference» element into е, has the form

(7)

We specify the space of functions and the set of degrees of freedom as follows:

(8)

The Lagrange basis in consists of the functions, where for i = 1,2,4 and j = 1 for i = 3 being the images of the basis (5) under the mapping (7) and satisfying the condition

(9)

Thus, the triple is a finite element that is affinely equivalent to the «reference» element [12].

Now denote the set of all nodes of the elements by and number them from 1 to With each node we associate the number equal

to the number of degrees of freedom related to this node. Observe that is occurred when the node is the midpoint of the side of a triangular element being a part of the boundary and for all remaining nodes of

This is the global numbering of nodes. We also introduce the local numbering. The couple is assumed to be a local number of the node of an element. To any local number there corresponds one and only one global number hence, we can introduce the function so that In addition, for an element we denote the local analogue of the parameter by, i.e., for

At each node, , we specify numbers Construct the function defined on such that

(10)

By the construction, the function is uniquely defined on each. Put

Then for any

Lemma 2. The function defined by the relation (10) belongs to

Proof. The BFS-rectangles are elements of the class i.e., the function and its first-order derivatives are continuous on the sides common for two elements of this type [10].

Now let be an arbitrary triangular cell and be a rectangular cell that has a common side with (Figure 4). Because of construction, the values of the nodal parameters of the functions and on are equal. In addition, the traces of these functions and their first-order derivatives with respect to on are polynomials of degree 3 in and are uniquely defined by the sets of nodal parameters related to the side

. Hence, the functions and coincide on

along with their first-order derivatives. ,

Corollary 1. [12,21]. Thus, we can define the finite element space as follows:

Let. Define its interpolant in the following way:

(11)

With the help of the Theorem 1, the following estimate can be proved in the usual way (see, for instance, [12,14]).

Theorem 2. Assume that And let be its interpolant defined by (11). Then

Here and later constants are independent of and

5. Numerical Example

We illustrate properties of the proposed finite elements by the following example. Let be a right triangle with unit catheti (see Figure 5) and a boundary Consider the problem

with the right-hand side

It has the exact solution

Subdivide the domain into elementary squares (with triangles adjacent to the hupotenuse) by drawing two families of parallel straight lines and with mesh size

To compare accuracy with decreasing mesh size, we construct a system of linear algebraic equations of the finite element method with the BFS-elements on the elementary squares and the proposed elements on the

triangles for Since the exact solution is known, the difference between exact and approximate solutions can be expressed in the explicit form. As a result, we have the following accuracy.

Theoretically, in the finite element method we have the estimate [12,14]

Combining it with the estimate in Theorem 2, we arrive at the following error estimate for an approximate solution:

(12)

Comparing the last two results in Table 1, observe that they are close to the asymptotic values 4 and 3, respectively.

6. Summary and Further Implementations

Thus, the use of the proposed triangular finite elements only near a boundary extends the field of application of the BFS-elements at least for second order equations. In its turn, an approximate solution is of the class enabling one to calculate a residual directly and considerably simplifies a posteriori accuracy estimates for an approximate solution.

In principle, to achieve the same order accuracy, instead of the BFS-elements, one can use the Lagrange bicubic elements on rectangles and the Lagrange elements of degree three on triangles. But in this case a general advantage of Hermite finite elements in comparison with Lagrange ones makes itself evident in the number of unknowns of a discrete system. In particular, Table 2 shows the number of degrees of freedom for an approximate solution which is equal to the number of unknowns and the number of equations in a discrete system of linear algebraic equations for the example from the above section.

Observing a considerable gain in the number of unknowns, with the further refinement of triangulation, the number of unknowns (and equations) asymptotically tends to the ratio of 4:9 in favour of Hermite elements.

Now we notice an apparent inconvenience and show a way to overcome it that is also useful for nonconforming grid refinement. The triangulation shown in Figure 1 is constructed by adjusting cells of a rectangular grid in the - and -directions. It imposes restrictions on the ratio of steps for a rectangular grid inside a domain. We show that the proposed triangular elements are sufficient to construct a conforming interpolant, generally speaking, on an “uncoordinated” grid without restriction on the ratio between steps.

Considering a typical case shown in Figure 6, the node is a so-called “hanging” node. At the algorithmic level, the challenge is solved as follows. For the degrees of freedom at the node b, instead of the corresponding variational equations for an approximate solution we write 4 linear algebraic equalities which express the quantities and in terms of 16 degrees of freedom of the neighbouring rectangle This trick ensures interelemental continuous differentiability of an interpolant between and Thus in this case one again may implement proposed pair of elements in the frame of the conforming finite element method with estimate (12).

REFERENCES