Abstract
In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.
1. Introduction
The main concern of this work consists of finding, using the Finite Element Method (FEM), the numerical solution of differential equations in which integrable Henstock–Kurzweil functions defined on the interval appear. We will say simply function. The space of these functions is denoted by . In particular, we consider functions that are highly oscillatory near the singularity (no Lebesgue integrable). For that, some results that guarantee the applications of the FEM are necessary. As the first step, conditions that guarantee that the product of functions be an function. This is a consequence of the weak formulation of the differential equation. As the second step, numerical methods of integration for functions must be used, in particular, for the case of the highly oscillatory functions. The trapezoid and Simpson methods are commonly used for the numerical calculation of integrals. However, when the function is highly oscillatory with a singularity, the performance of these methods can diminish and for the case of functions, fail in the calculation. For that reason, it is necessary to use quadratures which can be capable to handle these difficulties. Different quadrature methods have been developed and improved for calculating the integral of functions that belong to different spaces. Some examples of these methods are the Gauss–Raudi [1], Gauss–Legendre and Gauss–Lobatto [2], Newton–Cotes (open and closed) and first kind of Gauss–Chebyshev quadrature rules [3]. In particular, the authors of [2] proposed a numerical improvement to the Gauss–Lobatto quadrature. The quadratures used in this work are the Lobatto quadrature [1, 2, 4] and the open quadrature defined for functions with a singularity [5]. The results obtained are compared with those given by the trapezoid quadrature. These three quadrature methods are described below.
Different authors have been studied differential equations for functions. In [6], considered the Schrödinger equation involving the Henstock–Kurzweil integral. The author proves the existence and uniqueness of the initial value problem for the Schrödinger equation when the function of the right side of the equation is . It is proved that the solution of the equation and its derivative belong to space [7]. One example is shown to illustrate the application of these results, in which the solution of the problem presented belongs to and this example is not covered by any result using the Lebesgue integral. In [8], the Laplace transform for functions is considered on appropriated spaces to guarantee its existence, continuity and differentiation but not examples of applications were presented in this work. In [9], the Henstock–Kurzweil integral is considered from a distributional analysis and two fixed-point theorems are presented.
This work is organized as follows: In Section 2, the basic elements of the FEM are given; in Section 3, the definition of the function and some basic results, which allow the application of the FEM, are given; in Section 4, some quadratures for functions are described; in Section 5, numerical examples are presented in order to validate the proposed methodology. Finally, in Section 6, the results and perspectives of the work are discussed.
2. Preliminars
The numerical solution of differential equations has great importance in mathematics and engineering since they appear in many applications. There are different methods for the numerical solution of differential equations such as finite differences, FEM, finite volume method. We are interested in the FEM due to the weak formulation of this method, that is, the method uses a variational problem associated to the differential equation. Since the modeling of many problems is made through the variational formulation, the application of the FEM is natural.
2.1. Weak Formulation
To illustrate the weak formulation of elliptic boundary problems and its resolution by the FEM, we consider the following problem:
Find such that
where the set of admissible functions is defined by
The Equation (1) appears in some physical processes, for example, heat conduction or convection on a flat wall or on a bar, flow-through channels or pipes, axial deformation of bars, among others.
Note: In the analysis carried out in this work, generality is not lost if we consider .
The weak formulation of the elliptic problem (1)–(3) is given by
The variational problem (5) is also known as the weak form or Galerkin form of the differential equation. The linear space is approximated by the discrete space (), defined by
where are called base functions. For more details, see [10, 11].
2.2. Finite Element Method
The FEM provides a technique in which the domain is represented as a geometrical set of simple domains, that are called finite elements, which leads to a derivation of approximate functions on each element, usually algebraic polynomials that are chosen based on the characteristics of the problem that is being analyzed. The polynomials of degree one are commonly used because they are mathematically simple and easy to implement computationally.
Then, we can approximate the variational problem (5) for the following discrete variational problem:
Find such that
with where denote .
Substituting in (7), we get that the discrete variational problem is equivalent to calculate such that
which is reduced to solve a system of linear equations, where the matrix of the problem is symmetric and positive definite. The integrals in (8) can be calculated using the trapezoid or Simpson rule, when the diffusion parameter and the source , are functions with finite energy (square integrable), that is, and . These classical methods of integration can not be applied for functions in [5], therefore it is necessary to apply special quadratures for solving these integrals numerically and then to apply the finite element method for .
3. The Henstock–Kurzweil Integral
In this section we will present some of the most elementary and important properties of the Henstock–Kurzweil integral; specifically, we will mention some algebraic properties.(i)Two compact subintervals are called nonoverlapping if . The length of an interval , with , is defined as(ii)A subpartition of is a finite collection of nonoverlapping compact subintervals of with . A partition of is a subpartition such that .(iii)A tagged partition (tagged subpartition) of is a set of ordered pairs such that the collection is a partition (subpartition) of and , where the point is called tag associated to the subinterval , for every . We denoted by , the set of all tagged partitions of .(iv)A function is called a gauge on if , for every .(v)Given a gauge on and a tagged partition (subpartition) of , we say that is -fine if
for every .
Definition 1. A function is Henstock–Kurzweil integrable on (briefly, integrable) and is its integral if for every there exists a gauge on such that for every -fine tagged partition of the inequality holds.
The integral is unique as a consequence of Cousin’s Lemma. We denote the space of all Henstock–Kurzweil integrable functions by and to the value of the integral by .
The integral is not an absolute integral in the sense that if is integrable on , the function is not necessarily integrable on (see [7]). However, if is integrable we have the following:
Theorem 2. If are integrable functions on , then
The following result characterizes the integrability of a function when there is no particular value that can be predicted as the value of the integral, or it is unknown.
Theorem 3 (Cauchy’s criterion). A function is integrable on if and only if for every there exists a gauge on such that for every -fine tagged partitions and of the inequality holds.
Given the relationship between the concept of measurability and integrability, as in the Lebesgue integral, we cannot ignore this concept of great relevance in this work, so in this section we will present some important results related to measurability.
Definition 4. A function is called simple on if there is a finite sequence , of measurable sets such that for and where for , , i.e. is constant on the measurable set .
Definition 5. A function is Lebesgue measurable on if and only if there exists a sequence of simple functions on such that
Theorem 6. If is integrable on , then is Lebesgue measurable on .
Definition 7. Let be a function. The variation of on is
The function is of bounded variation on if . The space of all bounded variation functions is denoted by .
Definition 8. A function satisfies the Lipschitz condition if there exists such that , , and is of bounded slope variation (briefly, BSV) on ifis bounded for all divisions .
The basic functions of the MEF belong to . This fact, together with the following Theorem, guarantees that the product of functions that appear when applying the MEF is integrable, [7].
Theorem 9 (of Multiplication). If and , then .
Observe that in the variational formulation (5), we required the derivative of the solution , the derivatives of the test functions and the integrability of the product of these derivatives with the function . In [12] p. 11, an example that this product do not belongs to is presented. By to guarantee the existence of those integrals, the following Theorem will be used, [7].
Theorem 10. A function is the primitive of a function of bounded variation on if and only if satisfies the Lipschitz condition and is of bounded slope variation on .
4. Numerical Integration for Some Henstock–Kurzweil Integrable Functions in One Dimension
The numerical integration arises from the difficulty and/or impossibility of solving analytically some integrals that are of interest to the sciences. These difficulties can be due to the integration domain, the complexity of the functions, or both. Thus, the integral of a function is approximated as a finite sum of the area of rectangles of height and width ,
which is similar to the definition of Riemann [1, 12]. To calculate the integral numerically, we need a partition of the interval of subintervals and then make the sum of the values of the function with a weight . If the function is integrable in some sense, it is expected that the sum of the right hand of (17) converge to the integral when . Different weights and points (called nodes) generate different integration methods [1]. Furthermore, when grows, also the precision of the algorithm grows, except for rounding errors. In fact, the accuracy of the approximation depends strongly on the type of function that we want to integrate, for example, if the function presents some singularity, it is necessary to remove it or split the interval into subintervals to avoid adding the singularity.
In the following, we consider three quadrature methods. The first is the trapezoid method, which is one of the Newton-Cotes rules more used in practice.
Trapezoid method. This quadrature is well-known and it is given by
and the corresponding error is obtained from where . The trapezoid method can be applied to a partition of the interval .
As we have mentioned previously, classical integration methods can not be applied directly. The trapezoid method is not enough to solve the integral with functions. In what follows, we show three quadratures to calculate the numerical integrals of functions in one dimension.
Open quadrature. It is denoted by , and it is defined as follows, [5],
This quadrature is used specially for functions with a singularity in in the interval since it avoids the singularity in that point.
Close quadrature. The approximation is computed as follows
this quadrature is used to solve the integral in . Where, in (19) and (20), is defined by
Gauss–Lobatto. This quadrature is given by, [2],
which is exact for polynomials of degree at most , nodes , weights and variables and can be determined by the undetermined coefficients method.
The following result about the integration of highly oscillatory functions near of the singularity, can be found in [5].
Theorem 11. Let and in , where . If converges, then is integrable in and
5. Numerical Results
In this section, we present some numerical results obtained with the methodology proposed in this work. We will show how the classic methods of integration are not enough to solve elliptical differential equations with -functions. To test the convergence of the numerical results, different meshes are considered for the finite element discretization of the elliptical problems.
In order to present the numerical results, we consider the following elliptic problem:
Find such that
Also, we consider the domain , i.e., .
Example 1. For this first example, we discretize the domain with 100 nodes and 99 elements, i.e., we consider 0.0101 as the discretization parameter. We consider the diffusion parameter and the source as highly oscillating functions, given by and , respectively, with and . The exact solution of this elliptical problem is given by .
The numerical results are summarized in Table 1, and the following relative errors are included:
These results are obtained by the FEM, where the integrals that appear in (8) are solved using the integration methods seen in the previous Section. In addition, we observe that when increases the relative errors increase too, therefore it will be necessary to use finer meshes. We note that the trapezoid rule in not enough to obtain reasonable results, thus we have to apply special quadratures for this type of functions. Figure 1 shows the diffusion parameter in two different meshes, namely, a coarse one () and fine one (), whereas Figure 2 shows the exact solution and the approximated solutions using different quadratures, with .
(a)
(b)
Table 2 shows the relative errors for two mesh refinement. In this Table, we observe that using special quadratures for highly oscillating functions, it is possible to obtain precision. Figure 3 shows the diffusion parameter for the two meshes, with .
(a)
(b)
Example 2. Now, we consider the following diffusion parameter , which is a continuous function defined by and the source is given byThe exact solution of the differential equation (24) and (25) is .
The corresponding numerical results have been summarized in Table 3, these results show convergence with the refined meshes. As in the previous example, the trapezoidal rule is not applicable for this kind of functions unless we use very refined meshes, which is very expensive computationally speaking. The diffusion parameter and the approximate solutions , for 0.002501, are shown in the Figures 4 and 5, respectively.
Example 3. We consider the following highly oscillatory functions with singularity in , and . The exact solution of this problem, (24) and (25), is given by . To discretize , we will use 0.01, and two refinements, namely 0.005 and 0.0025.
In Table 4, the relative errors and the order of convergence obtained by three different meshes to solve the elliptic problem are shown. In this case, the trapezoid and close quadratures cannot be applicable for this kind of functions. Also, we note that the open quadrature, , has better convergence. In addition, for the first mesh refinement, from to , we obtain convergence of second order, and convergence superlinear, for and , respectively. Figure 6 shows the diffusion parameter and the approximate solution , for .
(a)
(b)
Example 4. In this example, we use the constant diffusion parameter and the following highly oscillatory source . The exact solution of the elliptical problem (24) and (25) is also highly oscillatory and it is given by , with and . To discretize we used , and two refinement of the mesh, i.e., and .
The relative errors for three different meshes and different quadratures, for the case are summarized in Table 5. In order to apply the trapezoid method is necessary to use finer meshes, which is very expensive computationally speaking.
The numerical results reported in Table 6 have been obtained taking different values of and . This table shows that our proposed methodology works. The exact solution and the approximate solutions, where and , are shown in Figure 6. Again, we can note that the trapezoid method does not work for the highly oscillatory functions.
Example 5. For the final experiment, unlike the previous ones, let us consider homogeneous Cauchy and Neumann boundary conditions, the constant diffusion parameter and the source , that is, the only -function for this differential equation is .The exact solution of this boundary problem is given by .
The results for different meshes are summarized in Table 7. In this case, the results obtained with the trapezoid quadrature are as good as the results obtained by close, open and Lobatto quadratures.
6. Conclusions
The Finite Element Method (FEM) has been used to solve differential equations when the functions involved are continuous, or they are square-integrable. In this work, the FEM was used to solve elliptical problems where the functions involved are integrable. The numerical results, developed in the numerical examples, have shown the feasibility of FEM when special quadratures to solve these problems are used. The open quadrature and the Lobatto quadratures has shown good results. The existence and uniqueness are not studied for the problems considered in this work. These points, as well as the application to other types of functions and other types of differential equations, will be studied in future works.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors want to acknowledge the financial support from PRODEP through a postdoctoral scholarship for the first author. The authors acknowledge also the support from the Math Graduate program at Benemérita Universidad Autónoma de Puebla. Finally, special thanks are due to the anonymous reviewers for their valuable advice and suggestions.