The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle

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.


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][13][14][15][16][17][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   1 C  of functions which are continuously differentiable on the closure  of the domain. . Assume that we can construct a triangulation h 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 h 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 h .On rectangular cells we use the Bogner-Fox-Schmit element [1].It is defined by the triple , ,  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] e e e h a a 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 for , , , 1, , 4, 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 x a h y a h x a h y a h ˆˆ, e e e e e x a h y a h ˆˆ, e e e e e x a h y a h ˆˆ, e e e e a x h y a h ˆˆ, e e e e a x h y a h ˆˆ, e e e e a x h y a h ˆˆ, e e e e a x h y a h x a h a y h x a h a y h x a h a y h x a h a y h ˆˆ, e e e e a x h a y h ˆˆ.
e e e e a x h a y h
We define the space of functions and the set  of degrees of freedom as follows: . Observe that at each of the nodes , and there are 4 degrees of freedom and at the is only one degree.Besides, the degrees om the nodes , and coincide with nodes of the FS-elem Lemma 1.The triple The direct calculations show that the Lagrange basis has the following form: Let be an arbitrary function.Along the side polynomial of degree 3 in ˆv a  and of nodal parameters.On the side similar statements are valid within the replacement of and the finite norm .
We also use the seminorm , .
consist uncti degrees of freedom : .
By the construction, the function is uniquely defined on each for any .

  2 h
Thus, we can define the finite element space as fols: : .
With the help of the Theorem 1, the following estimate can be proved in the usual way (see, for instance, [12,14]). Theorem Here and later constants are independent of and

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 a boundary

triangles for
Since the exact solution is known, the  betw xact and appressed in the explicit form.ollowing accuracy.Theoretically, in the finite element method we ha e the estim een e proximate solutions can be e As a result, we have the f x v ate [12,14] 3 , , , Combining it with the estimate in Theorem 2, we arrive at the following error estimate for an approximate solution: Comparing the last two results in Table 1, observe that they are close to the asymptotic values 4 and 3, respectively.

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   2 H  onsid-enabling one to calculate a residual directly and c erably simplifies a posteriori accuracy estimates for an approximate solution.
In principle, to achieve the same order a stead range nts on rectang n tria which is equal to the number of unknowns and the number of equations in a discrete system le 1. Accuracy of an approximate solution.
ccuracy, inof the BFS-elements, one can use the Lag e bicubic elem les and the Lagrange elements of degree three o ngles.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 of linear algebraic equations for the example from the above section.Considering a typical case shown in Fi , the no is a so-called "hanging" node.the rithm c level, the challenge is solved as follows.For the degr of freedom at the node b, instead of the correspond variational equations for an approximate solution we write 4 linear algebraic equalities which quantities poses rest on the ratio of steps for a rectangular grid inside a domain.We show that the proposed triangular elements are sufficient to construct a conform rdinated" grid with t restriction on the

Figure 1
Figure 1.A subdivision of a domain.2. Triangulation of a Domain and the Bogner-Fox-Schmit Element Let be a convex polygon with a boundary 2 R  .  Assume that we can construct a triangulation h 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 h 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
of two meshes.Hence, this feature has no influence on interelemental continuity inside the domain.properties, we use the usual tations for Sobolev spaces.Here   0 H  is the Hilbert space of functions, Lebesgue measurable on ,  equipped with the inn oduct er pr Figure 3.A triangular cell

1 Figure 4 .
Figure 4. Two neighbouring e ments of different types.
Figure 6.A "hanging" node b.Obs f unknowns , the number of unknowns (an equations) asymptotically tends to the ratio vour of Herm ts.Now we notice nt inconvenie w vercome lso useful fo grid re ent.Th triangulation sh n in Figure constructed by sting cells ectangular in the erving a considerable gain in the number o , with the further refinement of triangulation d of 4:9 in fa an appare ite elemen nce and show a ing ay to o it that is a r nonconform finem e ow 1 is adju of a r grid 1 x -and 2 x -directions.It im rictions This trick ensures intereledifferentiability of an interpolantThus in this case one again of elements in the frame of , method with estimate (12). .