Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems

Abstract

One-dimensional fourth-order boundary value problems (BVPs) play a critical role in engineering applications, particularly in the analysis of beams. Current numerical investigations primarily concentrate on homogeneous boundary conditions. In addition to its high precision advantages, the Hermite finite element method (HFEM) is capable of directly computing both the function value and its derivatives. In this paper, both the cubic and quintic HFEM are employed to address two prevalent non-homogeneous fourth-order BVPs. Furthermore, a priori error estimations are established for both BVPs, demonstrating the optimal error convergence order in $H^2$ semi-norm and $L^2$ norm. Finally, a numerical simulation is presented to validate the theoretical results.

1. Introduction

In the analysis of the bending behavior in plates and beams, fourth-order partial differential equations are commonly employed to describe their deflection, such as the fourth-order beam equation or the biharmonic equation [1]. These equations are widely utilized for modeling the bending response of beams and plates under various loading conditions. Additionally, fourth-order equations play a significant role in representing the behavior of thin membranes subjected to different loading conditions [2,3]. The utilization of one-dimensional fourth-order equations is crucial for accurately modeling the behavior of diverse physical systems, particularly those involving thin structures [4].

Numerical methods for solving one-dimensional fourth-order equations, such as the biharmonic equation, often involve discretization schemes that can handle the high-order derivatives. One common approach is to use finite difference methods [5], spectral methods [6] or finite element methods [7]. In finite difference methods, the fourth-order equation is typically discretized by using central difference approximations for the spatial derivatives, this discretization leads to a system of linear equations that can be solved using standard linear algebra techniques [8]. The spectral methods are based on approximating the unknown function by using a sum of basis functions, such as Fourier or Chebyshev polynomials, which can accurately represent high-order derivatives [9]. The biharmonic equation can be solved by transforming it into a system of algebraic equations in the spectral domain. The finite element methods can also be used to solve fourth-order equations [10,11]. In this approach, the domain is discretized into finite elements, and the weak form of the equation is derived. The unknown function is approximated within each element by using shape functions, and a system of algebraic equations is obtained by applying the Galerkin method. Comparing to the finite difference methods and spectral methods. The finite element method is suitable for a variety of complex geometry and boundary conditions, including unstructured grids and irregular boundaries, so it is more flexible when dealing with complex problems. Among the finite element method, HFEM has high numerical accuracy, especially in solving problems requiring high-order derivatives [12,13,14]. Due to its high accuracy, HFEM typically have higher convergence rates. Most of these existing numerical methods for the fourth-order model primarily focus on the fourth-order term and often overlook the general case. Furthermore, the majority of studies are confined to homogeneous boundary conditions.

The Hermite finite element method holds significant importance in solving beam problems due to its ability to provide highly accurate solutions. Unlike traditional finite element methods that approximate the solution using piecewise polynomials, the Hermite finite element method directly calculates both the function values and their derivatives at the element nodes. This direct calculation of derivatives enables the method to accurately capture the curvature and bending moments in beams, making it particularly well suited for analyzing beam structures. Additionally, the Hermite finite element method is known for its efficiency in handling higher-order differential equations, such as those governing beam bending, making it a valuable tool in the field of structural analysis and engineering. In this study, we explore the application of the Hermite finite element method to address general fourth-order boundary value problems, encompassing not only fourth-order terms but also second-order and reaction terms. This research distinguishes itself from previous studies on homogeneous boundary conditions by directly examining general non-homogeneous boundary conditions. We establish error estimations for both BVPs and demonstrate that the error estimates in $H^2$-seminorm and $L^2$-norm achieve optimal convergence order for $k=3$ and $k=5$.

The rest of this article is structured as follows. In Section 2, we present the two types of one-dimensional fourth-order BVPs and derive their weak formulation. In Section 3, we introduce the cubic and quintic HFEM and provide the corresponding finite element discretization scheme for the BVPs. In Section 4, we establish error estimates for both the $H^2$-seminorm and $L^2$-norm. In Section 5, we validate the results of theoretical analysis through numerical examples. Finally, in Section 6, we provide a summary of this paper.

2. One-Dimensional Fourth-Order Boundary Value Problems

In this section, we introduce the one-dimensional model for fourth-order boundary value problems along with its corresponding weak formulation.

2.1. Preliminaries

We use the standard notation for Sobolev spaces [15] and norms. The space $L^2\left(I\right)$ refers to functions on interval I that are square-integrable, with the inner product denoted by $(·,·)$ and a corresponding norm $\parallel ·\parallel$. For a non-negative integer k, we define $H^k\left(I\right)$ as the Hilbert space possessing weak $L^2$ derivatives up to order k, equipped with the norm ${\parallel ·\parallel }_k$ and seminorm ${|·|}_k$. When $I=[a,b]$, we use the notation $H_0^1\left(I\right)$ and $H_0^2\left(I\right)$ to denote specific spaces.

$$H_0^1\left(I\right)=\{v|v\in H^1\left(I\right),v\left(a\right)=0,v\left(b\right)=0\}.$$

$$H_0^2\left(I\right)=\{v|v\in H^2\left(I\right),v\left(a\right)=0,v\left(b\right)=0,v^{\prime }\left(a\right)=0,v^{\prime }\left(b\right)=0\}.$$

2.2. Problem Model

Consider the following one-dimensional fourth-order problem:

$$\frac{d^2}{d{x}^2}\left(c\left(x\right)\frac{d^{2}u\left(x\right)}{d{x}^2}\right)-\frac{d}{dx}\left(p\left(x\right)\frac{du\left(x\right)}{dx}\right)+q\left(x\right)u\left(x\right)=f\left(x\right),x\in I=(a,b),$$

(1)

where $c_{max}\ge c\left(x\right)\ge c_{min}>0$ and $p_{max}\ge p\left(x\right)\ge 0, q_{max}\ge q\left(x\right)\ge 0$ for all $x\in [a,b]$. With the following two common types of boundary conditions,

$$\begin{array}{c}\hfill u\left(a\right)={\alpha }_1,u\left(b\right)={\alpha }_2,u^{\prime }\left(a\right)={\beta }_1,u^{\prime }\left(b\right)={\beta }_2,\end{array}$$

(2)

$$\begin{array}{c}\hfill u\left(a\right)={\alpha }_1,u\left(b\right)={\alpha }_2,u^{″}\left(a\right)={\nu }_1,u^{″}\left(b\right)={\nu }_2.\end{array}$$

(3)

Remark 1.

When the Equation (1) degenerates to only terms of fourth-order, it becomes the Euler–Bernoulli beam equation.

2.3. Weak Formulation

In this subsection, we establish the weak formulation of the model by incorporating the boundaries (2) and (3).

Firstly, we derive the weak formulation of the boundary value problem by considering the fourth-order model (1) in conjunction with boundary conditions (2). First, by multiplying both sides of Equation (1) with the test function $v\left(x\right)\in H_0^2\left(I\right)$ and integrating over the interval $I=[a,b]$, while employing integration by parts, we obtain

$$\begin{array}{cc}\hfill {\int }_a^{b}fvdx& ={\int }_a^b\left({\left(c{u}^{″}\right)}^{″}-{\left(p{u}^{\prime }\right)}^{\prime }+qu\right)vdx\hfill \\ \hfill & ={\int }_a^b\left(-{\left(c{u}^{″}\right)}^{\prime }{v}^{\prime }+\left(p{u}^{\prime }\right)v^{\prime }+quv\right)dx+{\left(c{u}^{″}\right)}^{\prime }v{{|}_a^b-p{u}^{\prime }v|}_a^b\hfill \\ \hfill & ={\int }_a^b\left(c{u}^{″}{v}^{″}+p{u}^{\prime }{v}^{\prime }+quv\right)dx-c{u}^{″}{v}^{\prime }{|}_a^b\hfill \\ \hfill & ={\int }_a^b\left(c{u}^{″}{v}^{″}+p{u}^{\prime }{v}^{\prime }+quv\right)dx.\hfill \end{array}$$

Then, the variational formulation of (1) and (2) can be expressed as follows.

Problem${\mathrm{P}}^1$ Find $u\in V_1:=\{v|v\in H^2\left(I\right),v\left(a\right)={\alpha }_1,v\left(b\right)={\alpha }_2,v^{\prime }\left(a\right)={\beta }_1,v^{\prime }\left(b\right)={\beta }_2\}$, such that

$$A(u,v)={(f,v)}_1\forall v\in H_0^2\left(I\right),$$

(4)

where

$$A(u,v)={\int }_a^b\left(c{u}^{″}{v}^{″}+p{u}^{\prime }{v}^{\prime }+quv\right)dx,$$

(5)

$${(f,v)}_1={\int }_a^{b}fvdx.$$

Secondly, we obtain the weak formulation of the boundary value problem for the model (1) in conjunction with boundary conditions (3). Multiplying both sides of the equation by the test function $v\left(x\right)\in H_0^1\left(I\right)\cap H^2\left(I\right)$ and integrating over the interval I and using the integration by parts formula, we obtain

$$\begin{array}{cc}\hfill {\int }_a^{b}fvdx& ={\int }_a^b\left({\left(c{u}^{″}\right)}^{″}-{\left(p{u}^{\prime }\right)}^{\prime }+qu\right)vdx\hfill \\ \hfill & ={\int }_a^b\left(-{\left(c{u}^{\prime \prime }\right)}^{\prime }{v}^{\prime }+\left(p{u}^{\prime }\right)v^{\prime }+quv\right)dx+{\left(c{u}^{″}\right)}^{\prime }v{{|}_a^b-p{u}^{\prime }v|}_a^b\hfill \\ \hfill & ={\int }_a^b\left(c{u}^{″}{v}^{″}+p{u}^{\prime }{v}^{\prime }+quv\right)dx-c{u}^{″}{v}^{\prime }{|}_a^b\hfill \\ \hfill & ={\int }_a^b\left(c{u}^{″}{v}^{″}+p{u}^{\prime }{v}^{\prime }+quv\right)dx-c\left(b\right)u^{″}\left(b\right)v^{\prime }\left(b\right)+c\left(a\right)u^{″}\left(a\right)v^{\prime }\left(a\right).\hfill \end{array}$$

Then, the variational formulation of (1) and (3) reads

Problem${\mathrm{P}}^2$ Find $u\in V_2:=\{v|v\in H^2\left(I\right),v\left(a\right)={\alpha }_1,v\left(b\right)={\alpha }_2,v^{″}\left(a\right)={\nu }_1,v^{″}\left(b\right)={\nu }_2\}$, such that

$$A(u,v)={(f,v)}_2\forall v\in H_0^1\left(I\right)\cap H^2\left(I\right),$$

(6)

where $A(u,v)$ is defined as (5), and

$${(f,v)}_2={\int }_a^{b}fvdx+c\left(b\right)u^{″}\left(b\right)v^{\prime }\left(b\right)-c\left(a\right)u^{″}\left(a\right)v^{\prime }\left(a\right).$$

The bilinear form $A(u,v)$ satisfies the coercivity and continuity:

$$\begin{array}{cc}\hfill & A(u,v)\le C_1{|u|}_2{|v|}_2\forall u,v\in \hat{V},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & A(v,v)\ge C_2{|v|}_2^2\forall v\in \hat{V},\hfill \end{array}$$

(8)

where $\hat{V}$ represents $V_1$ or $V_2$ and the constants $C_1>0$, $C_2>0$. Note that by the Poincaré inequality [16], we know that ${|·|}_2$ is equivalent to ${\parallel ·\parallel }_2$, which illustrates that ${|·|}_2$ is also a norm for $\hat{V}$. By (7) and (8), from Theorem 11.2.2 in [17], Problem ${\mathrm{P}}^1$ and Problem ${\mathrm{P}}^2$ have a unique solution, respectively.

Throughout the paper, we utilize the symbol C to represent a general positive constant that is not influenced by specific quantities. For example, in the error analysis of numerical solutions, the constant C remains unaffected by spatial mesh-size h.

3. Finite Element Approximation

3.1. Hermite Finite Element Method

In this subsection, we present the Hermite finite element. Let us commence with the reference domain $\overline{\hat{I}}=[0,1],$ where $\hat{I}=(0,1)$. Subsequently, the local nodal basis functions were introduced on the aforementioned reference domain $\hat{I}$.

Firstly, we give the Hermite cubic basis functions. Let $N_1\left(\xi \right)$ be a cubic polynomial with $\xi \in \overline{\hat{I}}$:

$$N_1\left(\xi \right)=a_1{\xi }^3+b_1{\xi }^2+c_1\xi +d_1,$$

such that the following conditions are satisfied:

$$N_1\left(0\right)=1,N_1^{\prime }\left(0\right)=0,N_1\left(1\right)=0,N_1^{\prime }\left(1\right)=0.$$

Then, the coefficients of $N_1\left(\xi \right)$ can be determined by solving the following linear system:

$$\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 1& 1& 1& 1\\ 3& 2& 1& 0\end{array}\right)\left(\begin{array}{c}{a}_1\\ b_1\\ c_1\\ d_1\end{array}\right)=\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),$$

from which we have $a_1=2, b_1=-3, c_1=0, d_1=1.$ Thus,

$$N_1\left(\xi \right)=2{\xi }^3-3{\xi }^2+1.$$

(9)

Similarly, $N_2\left(\xi \right), N_3\left(\xi \right)$ and $N_4\left(\xi \right)$ can be found to have following formulas:

$$\begin{array}{c}{N}_2\left(\xi \right)=-2{\xi }^3+3{\xi }^2,\hfill \\ N_3\left(\xi \right)={\xi }^3-2{\xi }^2+\xi ,\hfill \\ N_4\left(\xi \right)={\xi }^3-{\xi }^2,\hfill \end{array}$$

(10)

where $N_2\left(\xi \right)$, $N_3\left(\xi \right)$, and $N_4\left(\xi \right)$ satisfy the following conditions individually.

$$N_2\left(0\right)=0, N_2^{\prime }\left(0\right)=0, N_2\left(1\right)=1, N_2^{\prime }\left(1\right)=0,$$

$$N_3\left(0\right)=0, N_3^{\prime }\left(0\right)=1, N_3\left(1\right)=0, N_3^{\prime }\left(1\right)=0,$$

$$N_4\left(0\right)=0, N_4^{\prime }\left(0\right)=0, N_4\left(1\right)=0, N_4^{\prime }\left(1\right)=1.$$

The plots of $N_i\left(\xi \right)$ for $i=1,2,3,4$ are depicted in Figure 1.

Secondly, we present the Hermite quintic basis functions. Denote $x_1=0, x_2=1$. Similarly, the formulas for $N_1\left(\xi \right),\cdots ,N_6\left(\xi \right)$ can be derived as follows:

$$\begin{array}{c}{N}_1\left(\xi \right)=-6{\xi }^5+15{\xi }^4-10{\xi }^3+1,\hfill \\ N_2\left(\xi \right)=6{\xi }^5-15{\xi }^4+10{\xi }^3,\hfill \\ N_3\left(\xi \right)=-3{\xi }^5+8{\xi }^4-6{\xi }^3+\xi ,\hfill \\ N_4\left(\xi \right)=-3{\xi }^5+7{\xi }^4-4{\xi }^3,\hfill \\ N_5\left(\xi \right)=-{\xi }^5/2+\left(3{\xi }^4\right)/2-\left(3{\xi }^3\right)/2+{\xi }^2/2,\hfill \\ N_6\left(\xi \right)={\xi }^5/2-{\xi }^4+{\xi }^3/2,\hfill \end{array}$$

(11)

where $N_1\left(\xi \right),\cdots ,N_6\left(\xi \right)$ satisfy the following conditions individually.

$$N_i\left(x_j\right)={\delta }_{ij}, N_i^{\prime }\left(x_j\right)=0, N_i^{″}\left(x_j\right)=0,i,j=1,2$$

$$N_i\left(x_j\right)=0, N_i^{\prime }\left(x_j\right)={\delta }_{i-2j}, N_i^{″}\left(x_j\right)=0i=3,4,j=1,2$$

$$N_i\left(x_j\right)=0, N_i^{\prime }\left(x_j\right)=0, N_i^{″}\left(x_j\right)={\delta }_{i-4j}i=5,6,j=1,2$$

where $\delta$ denotes the Kronecker symbol.

In general, we need to use the Hermite node basis in an arbitrary interval $[x_l,x_r].$ We transform the interval $[x_l,x_r]$ to the reference interval $[0,1]$. That is $\zeta :[x_l,x_r]\to [0,1]$ as

$$\zeta \left(x\right)=\frac{x-x_l}{x_r-x_l}.$$

(12)

This results in functions ${\psi }_i:[x_l,x_r]\to \mathbb{R}$ that are defined as follows.

• Hermite cubic basis function:

  $$\begin{array}{cc}\hfill & {\psi }_i\left(x\right)=N_i\left(\zeta \left(x\right)\right),i=1,2.\hfill \end{array}$$

  (13)

  $$\begin{array}{cc}\hfill & {\psi }_i\left(x\right)=(x_r-x_l)N_i\left(\zeta \left(x\right)\right),i=3,4.\hfill \end{array}$$

  (14)
• Hermite quintic basis function:

  $$\begin{array}{cc}\hfill & {\psi }_i\left(x\right)=N_i\left(\zeta \left(x\right)\right),i=1,2.\hfill \end{array}$$

  (15)

  $$\begin{array}{cc}\hfill & {\psi }_i\left(x\right)=(x_r-x_l)N_i\left(\zeta \left(x\right)\right),i=3,4.\hfill \end{array}$$

  (16)

  $$\begin{array}{cc}\hfill & {\psi }_i\left(x\right)={(x_r-x_l)}^2{N}_i\left(\zeta \left(x\right)\right),i=5,6.\hfill \end{array}$$

  (17)

3.2. Discrete Scheme

In this subsection, the finite element spaces for the boundary value problem will be discussed. We consider a uniform partitioning of the interval $[a,b]$ denoted by $I_h:a=x_0<x_1<\cdots x_N=b$, where each subinterval is represented as $I_n=[x_n,x_{n+1}]$ for $n=0,\cdots ,N-1$. The finite element space $V_h$ is defined as follows:

$$V_h=\{v_h\in H^2\left(I_n\right):v_h\in {\mathcal{P}}_k\left(I_n\right),k=3,5\}.$$

In the context of a given integer $k=3\mathrm{or}5$, ${\mathcal{P}}_k\left(I_n\right)$ represents the space of polynomials with a degree no greater than k.

Furthermore, we define the discrete bilinear form $A_h$ as

$$\begin{array}{c}\hfill A_h(u_h,v_h)=\sum _{I_n\in I_h}{\int }_{I_n}\left(c{u}_h^{″}{v}_h^{″}+p{u}_h^{\prime }{v}_h^{\prime }+q{u}_h{v}_h\right)dx\forall u_h,v_h\in V_h,\end{array}$$

(18)

and discrete norm as

$$\begin{array}{c}\hfill {\parallel v_h\parallel }_h^2=\sum _{I_n\in I_h}{|v_h|}_{2,I_n}^2.\end{array}$$

(19)

Below, we will present the discrete schemes for the initial boundary value problem (4) and (6). For the problem (4),

Problem${\mathrm{P}}_h^1$ Find $u_h\in V_{h1}:=\{v_h\in V_h:v_h\left(a\right)={\alpha }_1,v_h\left(b\right)={\alpha }_2,v_h^{\prime }\left(a\right)={\beta }_1,v_h^{\prime }\left(b\right)={\beta }_2\}$

$$A_h(u_h,v_h)={(f,v_h)}_1\forall v_h\in V_{h1}^0,$$

(20)

where $V_{h1}^0:=\{v_h\in V_h:v_h\left(a\right)=0,v_h\left(b\right)=0,v_h^{\prime }\left(a\right)=0,v_h^{\prime }\left(b\right)=0\}.$

And for the problem (6),

Problem${\mathrm{P}}_h^2$ Find $u_h\in V_{h2}:=\{v_h\in V_h:v_h\left(a\right)={\alpha }_1,v_h\left(b\right)={\alpha }_2,v_h^{″}\left(a\right)={\nu }_1,v_h^{″}\left(b\right)={\nu }_2\}$

$$A_h(u_h,v_h)={(f,v_h)}_2\forall v_h\in V_{h2}^0,$$

(21)

where $V_{h2}^0:=\{v_h\in V_h:v_h\left(a\right)=0,v_h\left(b\right)=0\}.$

Note that the bilinear form $A(u,v)$ satisfies the coercivity and continuity:

$$\begin{array}{cc}\hfill & A_h(u_h,v_h)\le M{\parallel u_h\parallel }_h{\parallel v_h\parallel }_h\forall u_h,v_h\in {\tilde{V}}_h,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & A_h(v_h,v_h)\ge \alpha {\parallel v_h\parallel }_h^2\forall v\in {\tilde{V}}_h,\hfill \end{array}$$

(23)

where ${\tilde{V}}_h$ represents $V_{h1}^0$ or $V_{h2}^0$ and the constants $M>0$, $\alpha >0$. By [18], we know that ${\parallel ·\parallel }_h$ is a norm for ${\tilde{V}}_h$.

Similarly, by (22) and (23), from Theorem 11.2.2 in [17], the discrete schemes Problem ${\mathrm{P}}_h^1$ and Problem ${\mathrm{P}}_h^2$ have a unique solution, respectively.

For simplicity, let us consider Hermite cubic basis function, i.e., $k=3$. We use the notation ${\psi }_{k,i}:[x_k,x_{k+1}]\to \mathbb{R}, i=1,2,3,4$. We define two global basis functions as follows.

For $k=0$, we define

$${\varphi }_0\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\notin [x_0,x_1]\hfill \\ {\psi }_{0,1}\hfill & \mathrm{if}x\in [x_0,x_1],\hfill \end{array}\right.$$

$${\varphi }_0^{\prime }\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\notin [x_0,x_1]\hfill \\ {\psi }_{0,3}\hfill & \mathrm{if}x\in [x_0,x_1].\hfill \end{array}\right.$$

For $k=1,2,\cdots ,N-2$, we define

$${\varphi }_k\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\notin [x_{k-1},x_{k+1}]\hfill \\ {\psi }_{k-1,2}\hfill & \mathrm{if}x\in [x_{k-1},x_k]\hfill \\ {\psi }_{k,1}\hfill & \mathrm{if}x\in [x_k,x_{k+1}],\hfill \end{array}\right.$$

$${\varphi }_k^{\prime }\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\notin [x_{k-1},x_{k+1}]\hfill \\ {\psi }_{k-1,4}\hfill & \mathrm{if}x\in [x_{k-1},x_k]\hfill \\ {\psi }_{k,3}\hfill & \mathrm{if}x\in [x_k,x_{k+1}].\hfill \end{array}\right.$$

For $k=N$, we define

$${\varphi }_N\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\notin [x_{N-1},x_N]\hfill \\ {\psi }_{N,2}\hfill & \mathrm{if}x\in [x_{N-1},x_N],\hfill \end{array}\right.$$

$${\varphi }_N^{\prime }\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\notin [x_{N-1},x_N]\hfill \\ {\psi }_{N,4}\hfill & \mathrm{if}x\in [x_{N-1},x_N].\hfill \end{array}\right.$$

Thus, the node function values and their derivatives for ${\varphi }_k\left(x_j\right)$ and ${\varphi }_k^{\prime }\left(x_j\right)$ for $k,j=0,1,\cdots ,N$ are given as follows:

$${\varphi }_i\left(x_j\right)={\delta }_{ij},{\varphi }_i^{\prime }\left(x_j\right)={\delta }_{ij}.$$

(24)

Finally, the Hermite cubic finite element space is defined as

$$S_h:=span\{{\varphi }_0,{\varphi }_0^{\prime },{\varphi }_1,{\varphi }_1^{\prime },\cdots ,{\varphi }_N,{\varphi }_N^{\prime }\}.$$

So for $u_h\in S_h\subset V_h$, we have

$$u_h\left(x\right)=\sum _{i=0}^N{u}_i{\varphi }_i\left(x\right)+u_i^{\prime }{\varphi }_i^{\prime }\left(x\right).$$

(25)

By (24), we obtain

$$u_h\left(x_i\right)=u_i,u_h^{\prime }\left(x_i\right)=u_i^{\prime },i=0,1,\cdots ,N.$$

(26)

Below we will give a discrete matrix representation of the problem (20) and (21). Let $u_h$ as defined in (25), and $v_h={\varphi }_j\left(x\right)\mathrm{and}{\varphi }_j^{\prime }\left(x\right)$ for $j=0,\cdots ,N$ in (20). We obtain

$$\left(\begin{array}{cccccc}{A}_h({\varphi }_0,{\varphi }_0)& ··& A_h({\varphi }_N,{\varphi }_0)& A_h({\varphi }_0^{\prime },{\varphi }_0)& ··& A_h({\varphi }_N^{\prime },{\varphi }_0)\\ :& \ddots & :& :& \ddots & :\\ A_h({\varphi }_0,{\varphi }_N)& ··& A_h({\varphi }_N,{\varphi }_N)& A_h({\varphi }_0^{\prime },{\varphi }_N)& ··& A_h({\varphi }_N^{\prime },{\varphi }_N)\\ A_h({\varphi }_0,{\varphi }_0^{\prime })& ··& A_h({\varphi }_N,{\varphi }_0^{\prime })& A_h({\varphi }_0^{\prime },{\varphi }_0^{\prime })& ··& A_h({\varphi }_N^{\prime },{\varphi }_0^{\prime })\\ :& \ddots & :& :& \ddots & :\\ A_h({\varphi }_0,{\varphi }_N^{\prime })& ··& A_h({\varphi }_N,{\varphi }_N^{\prime })& A_h({\varphi }_0^{\prime },{\varphi }_N^{\prime })& ··& A_h({\varphi }_N^{\prime },{\varphi }_N^{\prime })\end{array}\right)\left(\begin{array}{c}{u}_0\\ :\\ u_N\\ u_0^{\prime }\\ :\\ u_N^{\prime }\end{array}\right)=\left(\begin{array}{c}(f,{\varphi }_0)\\ :\\ (f,{\varphi }_N)\\ (f,{\varphi }_0^{\prime })\\ :\\ (f,{\varphi }_N^{\prime })\end{array}\right)$$

Simply denoted as $Ax=b$.

For simplicity of matrix representation, for $x\in [a,b]$, we denote by $u=u\left(x\right)$. Everywhere in what follows, when the meaning is clear, we do not indicate explicitly the dependence of various variables on x. Next, we can deal with the boundary conditions as follows:

For Problem ${\mathrm{P}}_h^1$, we can make the first row of matrix A become $(1,···,0)$, where the first element is 1 and the others are 0, and the $(N+1)$-th row of matrix A becomes $(0,···,1,···,0)$, where the $(N+1)$-th element is 1 and the others are 0, and the $(N+2)$-th row of matrix A becomes $(0,···,1,···,0)$, where the $(N+2)$-th element is 1 and the others are 0, and the $(2N+2)$-th row of matrix A becomes $(0,···,1)$, where the last element is 1 and the others are 0. Then, the first row of $b$ becomes ${\alpha }_1$, and the $(N+1)$-th row of $b$ becomes ${\alpha }_2$, and the $(N+2)$-th row of $b$ becomes ${\beta }_1$, and the $(2N+2)$-th row of $b$ becomes ${\beta }_2$.

For Problem ${\mathrm{P}}_h^2$, we perform a similar operation for matrix A and vector $b$. The difference is that the $(N+2)$-th row of $b$ becomes $(f,{\varphi }_0^{\prime })-c\left(a\right){\nu }_1$, and the $(2N+2)$-th row of $b$ becomes $(f,{\varphi }_N^{\prime })+c\left(b\right){\nu }_2$.

4. Error Estimations

In this section, we primarily establish error estimates in the $H^2$-seminorm and $L^2$-norm. To facilitate the subsequent proof process, we first present an interpolation estimate [17].

Lemma 1.

For every $I_n\in I_h$ and every $v\in H^s\left(I_n\right)$ with $2\le s\le k+1$, it holds that

$$\begin{array}{c}\hfill {\parallel v-v_I\parallel }_{m,I_n}\le C{h}^{s-m}{|v|}_{s,I_n}m=0,1,2.\end{array}$$

(27)

Theorem 1.

Let $u\in H^{k+1}\left(I\right)$ be the solution of Problem ${\mathrm{P}}^1$ or Problem ${\mathrm{P}}^2$ and $u_h$ be the solution of Problem ${\mathrm{P}}_h^1$ or Problem ${\mathrm{P}}_h^2$ , respectively, then we have

$$\begin{array}{c}\hfill {\parallel u-u_h\parallel }_h\le C{h}^{k-1}{|u|}_{k+1},\end{array}$$

(28)

where the constant C depends only on u and constants M.

Proof. 

By the Céa lemma, we have

$$\begin{array}{c}\hfill {\parallel u-u_h\parallel }_h\le \underset{v_h\in V_h}{inf}C{\parallel u-v_h\parallel }_h,\end{array}$$

and combining Lemma 1, we can obtain

$$\begin{array}{c}\hfill {\parallel u-u_h\parallel }_h\le C{h}^{k-1}{|u|}_{k+1}.\end{array}$$

This completes the proof. □

Theorem 2.

Let $u\in H^{k+1}\left(I\right)$ be the solution of Problem ${\mathrm{P}}^1$ or Problem ${\mathrm{P}}^2$ and $u_h$ be the solution of Problem ${\mathrm{P}}_h^1$ or Problem ${\mathrm{P}}_h^2$ , respectively, then

$$\begin{array}{c}\hfill {\parallel u-u_h\parallel }_0\le C{h}^{2k-2}{|u|}_{k+1},\end{array}$$

(29)

where the constant C depends only on u and constants M.

Proof. 

Setting $\psi \in V$ as the solution of the following variational problem

$$\begin{array}{c}\hfill A(v,\psi )=(u-u_h,v)\forall v\in V,\end{array}$$

(30)

where V represents $V_1$ or $V_2$.

From the above Equation (30), we know that $\psi$ is the solution of the adjoint problem Problem ${\mathrm{P}}^1$ or Problem ${\mathrm{P}}^2$ where the right hand side is set as $u-u_h$. Assume that ${\psi }_h\in V_h$ is the finite element solution of

$$\begin{array}{c}\hfill A_h(v_h,{\psi }_h)=(u-u_h,v_h)\forall v_h\in V_h,\end{array}$$

where $V_h$ is the finite element space corresponding to V.

Notice that

$$\begin{array}{c}\hfill (u-u_h,u-u_h)=A_h(u-u_h,\psi -{\psi }_h)+A_h(u_h,\psi )+A_h(u,{\psi }_h)-2A_h(u_h,{\psi }_h),\end{array}$$

(31)

where the last three terms of (31) $A_h(u_h,\psi )+A_h(u,{\psi }_h)-2A_h(u_h,{\psi }_h)$ can be rewritten as $A_h(u_h,\psi -{\psi }_h)+A_h(u-u_h,{\psi }_h)$, then by the Galerkin orthogonality, as $V_h\in V$, we can get

$$\begin{array}{c}\hfill A_h(u_h,\psi -{\psi }_h)+A_h(u-u_h,{\psi }_h)=0.\end{array}$$

Thus, (31) leads to

$$\begin{array}{c}\hfill (u-u_h,u-u_h)=A_h(u-u_h,\psi -{\psi }_h).\end{array}$$

(32)

By the continuity of $A_h$ and (28), we can obtain

$$\begin{array}{c}\hfill (u-u_h,u-u_h)\le M{\parallel u-u_h\parallel }_h{\parallel \psi -{\psi }_h\parallel }_h\le C{h}^{k-1}{|u|}_{k+1}{\parallel \psi -{\psi }_h\parallel }_h.\end{array}$$

By the results of the regularity theory of differential equations, we have

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_{k+1}\le C{\parallel u-u_h\parallel }_0.\end{array}$$

(33)

By interpolation estimation and (33), we have

$$\begin{array}{c}\hfill {\parallel \psi -{\psi }_h\parallel }_h\le C{h}^{k-1}{\parallel \psi \parallel }_{k+1}\le C{h}^{k-1}{\parallel u-u_h\parallel }_0.\end{array}$$

Thus, we get

$$\begin{array}{c}\hfill (u-u_h,u-u_h)\le C{h}^{2k-2}{|u|}_{k+1}{\parallel u-u_h\parallel }_0.\end{array}$$

Then,

$$\begin{array}{c}\hfill {\parallel u-u_h\parallel }_0\le C{h}^{2k-2}{|u|}_{k+1}.\end{array}$$

This completes the proof. □

Corollary 1.

From Theorems 1 and 2, we deduce that the $H^2$-seminorm error convergence orders for cubic and quintic HFEM are two and four, respectively, while the $L^2$-norm error convergence orders are four and six, respectively.

5. Numerical Simulations

In this section, we present numerical experiments conducted on one-dimensional fourth-order boundary value problems. For simplicity, let us make $c\left(x\right)$, $p\left(x\right)$, and $q\left(x\right)$ constants.

Example 1.

Following the example adopted from Khalifa and Aziz [19], we consider the beam of limited length on $[-1,1]$. Let $c\left(x\right)=1$, $p\left(x\right)=2$, and $q\left(x\right)=1$,

$$u^{\left(4\right)}\left(x\right)-2u^{\left(2\right)}\left(x\right)+u\left(x\right)=f\left(x\right),x\in (-1,1).$$

(34)

The exact solution of such equation is

$$u\left(x\right)=\frac{e^{1-x}cos\left(x\right)}{cos1}.$$

(35)

Bringing the exact solution (35) into (34) results in the corresponding right-hand term $f\left(x\right)$, along with the following two kinds of boundaries:

$$\left\{\begin{array}{c}u(-1)=e^2,u\left(1\right)=1,\hfill \\ u^{\prime }(-1)=e^2(tan1-1),u^{\prime }\left(1\right)=-(1+tan1).\hfill \end{array}\right.$$

(36)

and

$$\left\{\begin{array}{c}u(-1)=e^2,u\left(1\right)=1,\hfill \\ u^{″}(-1)=-2e^{2}tan1,u^{″}\left(1\right)=2tan1.\hfill \end{array}\right.$$

(37)

First, we aim to solve (20) and (21) using the cubic Hermite finite element method. In the subsequent numerical simulation, we employ uniform mesh generation. The $H^2$ semi-norm error $|u-u_h{|}_2$ and $L^2$ norm error $||u-u_h{||}_0$ and their convergence order are given in

Table 1. Numerical errors for cubic Hermite finite element method of (20).

Mesh Size (h)	${\mathit{H}}^2$ Semi-Error	Error Order	${\mathit{L}}^2$ Error	Error Order
1	1.1405		0.0499
1/2	0.2843	2.0043	0.0031	4.0035
1/4	0.0710	2.0017	1.9405 × 10$-4$	4.0042
1/8	0.0177	2.0004	1.2118 × 10$-5$	4.0013
1/16	0.0044	2.0001	7.5718 × 10$-7$	4.0003
1/32	0.0010	2.0000	4.7220 × 10$-8$	4.0031

and

Table 2. Numerical errors for cubic Hermite finite element method of (21).

Mesh Size (h)	${\mathit{H}}^2$ Semi-Error	Error Order	${\mathit{L}}^2$ Error	Error Order
1	1.1421		0.0378
1/2	0.2843	2.0062	0.0024	3.9537
1/4	0.0710	2.0019	1.5303 × 10$-4$	3.9945
1/8	0.0177	2.0004	9.5705 × 10$-6$	3.9990
1/16	0.0044	2.0001	5.9824 × 10$-7$	3.9998
1/32	0.0011	2.0000	3.7066 × 10$-8$	4.0125

. We can observe that the $H^2$ semi-norm error is second-order convergence and the $L^2$ norm error is fourth-order convergence, which agrees with the theoretical estimates (28) and (29) for $k=3$. In addition, Figure 2 shows the comparisons of the function images of the numerical solution and the true solution at the mesh size $h=1/8$; (a) shows the Problem ${\mathrm{P}}_h^1$, and (b) shows the Problem ${\mathrm{P}}_h^2$. From these figures, a good numerical approximation of the function and its derivatives can be observed.

Secondly, we use the quintic Hermite finite element method for numerical simulation. The numerical errors are given in

Table 3. Numerical errors for quintic Hermite finite element method of (20).

Mesh Size (h)	${\mathit{H}}^2$ Semi-Error	Error Order	${\mathit{L}}^2$ Error	Error Order
1	0.0081		7.7134 × 10$-5$
1/2	5.7698 × 10$-4$	3.8129	1.3077 × 10$-6$	5.8852
1/4	3.6226 × 10$-5$	3.9934	1.9897 × 10$-8$	6.0384
1/8	2.2574 × 10$-6$	4.0034	3.0660 × 10$-10$	6.0020
1/16	1.4090 × 10$-7$	4.0019	4.5808 × 10$-11$	2.7427

and

Table 4. Numerical errors for quintic Hermite finite element method of (21).

Mesh Size (h)	${\mathit{H}}^2$ Semi-Error	Error Order	${\mathit{L}}^2$ Error	Error Order
1	0.0081		7.7198 × 10$-5$
1/2	5.7698 × 10$-4$	3.8130	1.3078 × 10$-6$	5.8833
1/4	3.6226 × 10$-5$	3.9934	1.9897 × 10$-8$	6.0385
1/8	2.2574 × 10$-6$	4.0043	3.0662 × 10$-10$	6.0200
1/16	1.4090 × 10$-7$	4.0019	1.5879 × 10$-10$	0.9493

, from which we can observe that the $H^2$ semi-error reaches the convergence of order four and the $L^2$ error reaches the convergence of order six, which is consistent with the theoretical analysis. Furthermore, when compared to the calculation results of the cubic elements, the errors of the quintic elements are significantly smaller for the same grid. This demonstrates that higher order elements can enhance calculation accuracy. In addition, the approximation result of the numerical solution of the quintic Hermite finite element is given in Figure 3. It can be observed from (a) and (b) of Figure 3 that the values of the numerical solution, derivative and second-order derivative are close to the true solution.

6. Conclusions

It is widely acknowledged that fourth-order boundary value problems in one dimension play a crucial role in engineering, particularly in beam analysis. Current numerical research primarily focuses on addressing homogeneous boundary conditions. The Hermite finite element method is renowned for its precision advantages and its ability to directly calculate function values and their derivatives. In this paper, both the cubic and quintic HFEM are utilized to tackle two prevalent non-homogeneous fourth-order BVPs. Additionally, a priori error estimations are established for both BVPs, demonstrating optimal error convergence order in $H^2$ semi-norm and $L^2$ norm. Finally, a numerical simulation is presented to validate the theoretical results. For future research, we will explore alternative boundary conditions and 2D fourth-order boundary value problems. Furthermore, we can also extend our findings to general beam problems [20] and nonlinear ODEs [21].