Superconvergence of Some Output Functionals for Hermitian Finite Elements

I. Introduction

Traditional methods of solving equations of mathematical physics, such as finite element method, are to find a solution in the entire domain of definition. Corresponding methods of computational error analysis consist in defining the norm for difference between the exact and approximate solutions also in the whole domain. Meanwhile, in a number of applications, researchers are interested not in the solution as a whole, but only in its certain values or functionals named as goal-oriented or output ones. For example, in air flow around the body, engineers are interested in lift and drag, not in the solution at every point in space, especially away from the constructed body [1, 2]. In such cases, one would be interested in the calculation precision of these output functionals, and not of the entire solution. Moreover, with appropriate triangulation in the finite element method one can achieve a significant increase in the accuracy of the required functionals without increasing the computational time for the problem as a whole [2, 3].

One of the interesting tasks consists in computation of the solution values along the part of boundary with Neumann boundary condition. Such situation occurs when the flux (heat flux, for instance) at body surface equals to zero, and we need to know the resulting temperature at this surface.

In this article we consider a one-dimensional model problem with Neumann boundary condition at one of boundary points and try to find the solution value in the same point. For solving we use cubic Hermitian finite element method. Unexpectedly the ultrahigh order of convergence was obtained, which is not guaranteed by the finite element theory. Therefore the paper is devoted to the theoretical justification of this beneficial effect with the help of adjoint operators.

II. Formulations of problem

Consider one-dimensional model problem: find such that

(1)

(2)

Functions are smooth enough for further considerations. Here and after any constant is supposed to be positive and independent of meshsizes Introduce the functional space where last equality is valid in the sense of traces. We use the standard notation for functional spaces that can be found for instance in [4, 5].

To get the weak form of this problem, multiply equation (1) by arbitrary function and integrate by parts with the help of boundary conditions (2). As a result we get the equalities

(3)

In the weak form [4, 11] this problem is reformulated as follows:

(4)

Here and later is the bilinear form

are scalar product and norm in space

As it was mentioned above, in many real problems of mathematical physics the main important result is not the solution as a whole, but some its functionals. In this paper for above formulated problem we take the output functional

as an example and show that it is computed with better order of accuracy than the solution as a whole.

3. Finite element method with cubic Hermitian elements

First introduce mesh with nodes and notation

In accordance with handbooks [4, 11] introduce cubic basis functions of first type:

and second type:

Now introduce the finite element space consisting of piecewise polynomial functions of degree 3 as the linear span of these basis functions:

Due to Galerkin approach we get the following formulation:

(5)

Owing to basis functions in we have decomposition

with the vector of unknowns

where sign (…)^T means transposition. This vector is found from the system of linear algebraic equations [6, 10]

(6)

where is the global stiffness matrix andis the global right-hand side.

Introduce the error Subtracting (5) from (4) we get

(7)

This property is well-known as Galerkin orthogonality property.

The accuracy of approximate solution is known from handbooks [4, 11]:

(8)

where means the norms in the space When we have the discrete imbedding (inverse) inequality [4, 5]

(9)

Therefore we can guarantee only

(10)

4. A numerical result

Now consider the following concrete problem:

(11)

And then solve it by cubic finite element method that has been introduced before.

Table 1.

Numerical result for problem (11).

From this Table we can see that tends to zero asymptotically as but not as or Is this result just a fortuity for this simple problem?

Now let us prove it for more general problems.

5. The proof of the higher convergence order for this type of output functional

Now introduce the following adjoint problem with the output functional

(12)

For smooth coefficients this weak formulation is equivalent to the following problem:

Its sufficiently smooth coefficients imply a sufficiently smooth solution. Like for the original problem (4), Galerkin approach in this case gives the following weak formulation:

(13)

Therefore, we get the discrete system for adjoint problem

with the same matrix due to symmetry of bilinear form: After solving this system the vector of unknowns

defines the approximate solution

Theorem. For sufficiently smooth data of problem (1)-(2) we get

(14)

Proof. From (12), (7), (13) we get

(15)

By standard way we get inequality

Since both problems have sufficiently smooth solutions, then

It implies that

Thus, we indeed get the sixth order of convergence for this output functional.

6. Resume

By virtue of the duality theory we prove the higher order of convergence than that is following from standard theory of Hermitian finite element method.

Note, that this effect became possible for cubic finite elements. For linear ones we should not get higher order of accuracy. Indeed, repeating this way of justification we should get only

It is not unusual for this type of elements for which we have typical estimate

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