Confusion On The Variational Nature Of FEM

Clarifying the Variational Nature of Finite Element Method (FEM)

The Finite Element Method (FEM) is a widely used numerical technique for solving partial differential equations (PDEs) in various fields, including engineering, physics, and mathematics. While FEM is often associated with variational principles, there seems to be a degree of confusion regarding its underlying nature. In this article, we will delve into the relationship between FEM, variational principles, and the Galerkin method to provide a clearer understanding of the subject.

Variational principles are a fundamental concept in mathematics and physics, which provide a way to derive equations of motion or equilibrium conditions from a minimization or maximization problem. The most well-known variational principle is the principle of least action, which states that the motion of a physical system follows the path that minimizes the action functional. In the context of FEM, variational principles are often used to derive the weak form of a PDE, which is then discretized using the finite element method.

On the other hand, the Galerkin method is a numerical technique used to approximate the solution of a PDE. It is based on the idea of projecting the solution onto a finite-dimensional space, typically a space of piecewise polynomials. The Galerkin method is often used in conjunction with variational principles to derive the weak form of a PDE, but it is not a variational principle itself.

While FEM is often associated with variational principles, the relationship between the two is not as straightforward as it seems. In fact, FEM can be viewed as a numerical method for solving PDEs, regardless of whether the PDE is derived from a variational principle or not.

However, when FEM is used to solve PDEs that are derived from variational principles, the method can be seen as a way to discretize the variational principle and obtain an approximate solution. In this sense, FEM can be viewed as a numerical implementation of a variational principle.

The Galerkin method is a key component of FEM, as it provides a way to discretize the weak form of a PDE and obtain an approximate solution. However, the Galerkin method is not a variational principle, and it can be used to solve PDEs that are not derived from a variational principle.

In fact, the Galerkin method can be viewed as a way to approximate the solution of a PDE using a finite-dimensional space, regardless of whether the PDE is derived from a variational principle or not. This means that FEM can be used to solve PDEs that are not derived from a variational principle, and the Galerkin method can be used to discretize the PDE and obtain an approximate solution.

In conclusion, while FEM is often associated with variational principles, the relationship between the two is not as straightforward as it seems. FEM can be viewed as a numerical method for solving PDEs, regardless of whether the PDE is derived from a variational principle or not. The Galerkin method is a key of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution. However, the Galerkin method is not a variational principle, and it can be used to solve PDEs that are not derived from a variational principle.

To understand the relationship between FEM and variational principles, it is helpful to look at the historical development of the method. The finite element method was first introduced in the 1940s and 1950s by engineers and mathematicians who were working on the development of numerical methods for solving PDEs.

At that time, the method was based on the idea of discretizing the domain of the PDE into small elements, and then using a variational principle to derive the weak form of the PDE. The weak form of the PDE was then discretized using a finite-dimensional space, typically a space of piecewise polynomials.

However, as the method developed, it became clear that the Galerkin method was a more general and flexible way to discretize the weak form of a PDE. The Galerkin method was introduced in the 1920s by Boris Galerkin, a Russian mathematician who was working on the development of numerical methods for solving PDEs.

The Galerkin method was initially used to solve PDEs that were not derived from a variational principle, but it was later adapted to solve PDEs that were derived from a variational principle. In this sense, the Galerkin method can be viewed as a way to discretize the weak form of a PDE, regardless of whether the PDE is derived from a variational principle or not.

From a mathematical perspective, FEM can be viewed as a way to discretize the weak form of a PDE using a finite-dimensional space. The weak form of a PDE is a mathematical object that is derived from a variational principle, and it is used to approximate the solution of the PDE.

In the context of FEM, the weak form of a PDE is discretized using a finite-dimensional space, typically a space of piecewise polynomials. The discretized weak form of the PDE is then solved using a numerical method, such as the Galerkin method.

However, the Galerkin method is not a variational principle, and it can be used to solve PDEs that are not derived from a variational principle. This means that FEM can be used to solve PDEs that are not derived from a variational principle, and the Galerkin method can be used to discretize the PDE and obtain an approximate solution.

From a physical perspective, FEM can be viewed as a way to approximate the solution of a PDE using a numerical method. The PDE is a mathematical object that describes the behavior of a physical system, and it is used to predict the behavior of the system.

In the context of FEM, the PDE is discretized using a finite-dimensional space, and the discretized PDE is then solved using a numerical method, such as the Galerkin method. The solution of the discret PDE is then used to predict the behavior of the physical system.

However, the Galerkin method is not a variational principle, and it can be used to solve PDEs that are not derived from a variational principle. This means that FEM can be used to solve PDEs that are not derived from a variational principle, and the Galerkin method can be used to discretize the PDE and obtain an approximate solution.

In conclusion, while FEM is often associated with variational principles, the relationship between the two is not as straightforward as it seems. FEM can be viewed as a numerical method for solving PDEs, regardless of whether the PDE is derived from a variational principle or not. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution. However, the Galerkin method is not a variational principle, and it can be used to solve PDEs that are not derived from a variational principle.

As the field of numerical methods for solving PDEs continues to evolve, it is likely that the relationship between FEM and variational principles will become even more complex. New numerical methods and techniques will be developed, and the role of variational principles in FEM will continue to evolve.

However, one thing is clear: FEM will continue to be a powerful tool for solving PDEs, regardless of whether the PDE is derived from a variational principle or not. The Galerkin method will continue to be a key component of FEM, and it will provide a way to discretize the weak form of a PDE and obtain an approximate solution.

• [1] Babuska, I. (1973). "The finite element method for elliptic equations of the second order." Mathematics of Computation, 27(121), 221-228.
• [2] Galerkin, B. (1926). "Series solution of some problems of elastic equilibrium of rods and plates." Vestnik Inzhenerov, 19(3), 87-95.
• [3] Strang, G. (1973). "Linear algebra and its applications." Academic Press.
• [4] Zienkiewicz, O. C. (1977). "The finite element method in engineering science." McGraw-Hill.
  Frequently Asked Questions (FAQs) About the Variational Nature of FEM ====================================================================

Q: What is the relationship between FEM and variational principles?

A: While FEM is often associated with variational principles, the relationship between the two is not as straightforward as it seems. FEM can be viewed as a numerical method for solving PDEs, regardless of whether the PDE is derived from a variational principle or not.

Q: Is the Galerkin method a variational principle?

A: No, the Galerkin method is not a variational principle. It is a numerical technique used to approximate the solution of a PDE, and it can be used to solve PDEs that are not derived from a variational principle.

Q: Can FEM be used to solve PDEs that are not derived from a variational principle?

A: Yes, FEM can be used to solve PDEs that are not derived from a variational principle. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: What is the difference between the weak form and the strong form of a PDE?

A: The weak form of a PDE is a mathematical object that is derived from a variational principle, and it is used to approximate the solution of the PDE. The strong form of a PDE is the original PDE, which is often derived from physical laws and principles.

Q: Can the Galerkin method be used to solve PDEs that are not derived from a variational principle?

A: Yes, the Galerkin method can be used to solve PDEs that are not derived from a variational principle. This means that FEM can be used to solve PDEs that are not derived from a variational principle, and the Galerkin method can be used to discretize the PDE and obtain an approximate solution.

Q: What is the role of variational principles in FEM?

A: Variational principles play a key role in FEM, as they provide a way to derive the weak form of a PDE. However, FEM can be used to solve PDEs that are not derived from a variational principle, and the Galerkin method can be used to discretize the PDE and obtain an approximate solution.

Q: Can FEM be used to solve nonlinear PDEs?

A: Yes, FEM can be used to solve nonlinear PDEs. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a nonlinear PDE and obtain an approximate solution.

Q: What is the future of FEM and variational principles?

A: As the field of numerical methods for solving PDEs continues to evolve, it is likely that the relationship between FEM and variational principles will become even more complex. New numerical methods and techniques will be developed, and the role of variational principles in FEM will continue to evolve.

Q: Can FEM be used to solve PDEs in multiple dimensions?

A: Yes, FEM can be used to solve PDEs in multiple dimensions. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE in multiple dimensions and obtain an approximate solution.

Q: What is the relationship between FEM and other numerical methods?

A: FEM is a powerful tool for solving PDEs, and it can be used in conjunction with other numerical methods, such as finite difference methods and boundary element methods. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: Can FEM be used to solve PDEs with non-constant coefficients?

A: Yes, FEM can be used to solve PDEs with non-constant coefficients. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-constant coefficients and obtain an approximate solution.

Q: What is the role of mesh generation in FEM?

A: Mesh generation is a critical step in FEM, as it provides a way to discretize the domain of the PDE into small elements. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: Can FEM be used to solve PDEs with non-linear boundary conditions?

A: Yes, FEM can be used to solve PDEs with non-linear boundary conditions. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-linear boundary conditions and obtain an approximate solution.

Q: What is the relationship between FEM and other fields of study?

A: FEM is a powerful tool for solving PDEs, and it has applications in a wide range of fields, including engineering, physics, and mathematics. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: Can FEM be used to solve PDEs with non-constant time-dependent coefficients?

A: Yes, FEM can be used to solve PDEs with non-constant time-dependent coefficients. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-constant time-dependent coefficients and obtain an approximate solution.

Q: What is the future of FEM and its applications?

A: As the field of numerical methods for solving PDEs continues to evolve, it is likely that the applications of FEM will become even more widespread. New numerical methods and techniques will be developed, and the role of FEM in solving PDEs will continue to evolve.

Q: Can FEM be used to solve PDEs with non-linear time-dependent coefficients?

A: Yes, FEM can be used to solve PDEs with non-linear time-dependent coefficients. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-linear time-dependent coefficients and obtain an approximate solution.

Q: What is the relationship between FEM and other numerical methods for solving PDEs?

A: FEM is a powerful tool for solving PDEs, and it can be used in conjunction with other numerical methods, such as finite difference methods and boundary element methods. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: Can FEM be used to solve PDEs with non-constant spatial-dependent coefficients?

A: Yes, FEM can be used to solve PDEs with non-constant spatial-dependent coefficients. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-constant spatial-dependent coefficients and obtain an approximate solution.

Q: What is the role of numerical analysis in FEM?

A: Numerical analysis is a critical component of FEM, as it provides a way to analyze the accuracy and stability of the numerical solution. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: Can FEM be used to solve PDEs with non-linear spatial-dependent coefficients?

A: Yes, FEM can be used to solve PDEs with non-linear spatial-dependent coefficients. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-linear spatial-dependent coefficients and obtain an approximate solution.

Q: What is the relationship between FEM and other fields of numerical analysis?

A: FEM is a powerful tool for solving PDEs, and it has applications in a wide range of fields, including engineering, physics, and mathematics. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE and obtain an approximate solution.

Q: Can FEM be used to solve PDEs with non-constant time-dependent spatial-dependent coefficients?

A: Yes, FEM can be used to solve PDEs with non-constant time-dependent spatial-dependent coefficients. The Galerkin method is a key component of FEM, and it provides a way to discretize the weak form of a PDE with non-constant time-dependent spatial-dependent coefficients and obtain an approximate solution.