Explain the concept of numerical solutions of partial differential equations with finite element method.

The concept of numerical solutions of partial differential equations (PDEs) with the finite element method (FEM) involves approximating the solution of a PDE by dividing the domain into smaller subdomains, called finite elements.

In the FEM, the PDE is first transformed into a system of algebraic equations by discretizing the domain and approximating the solution within each finite element. This is done by using basis functions, which are typically polynomials, to represent the unknown solution within each element. The basis functions are chosen such that they satisfy certain properties, such as continuity and differentiability, and are often defined on a reference element.

The next step is to construct a global system of equations by assembling the local element equations. This is done by enforcing the continuity of the solution at the interfaces between adjacent elements. The resulting system of equations is typically a large sparse matrix equation.

To solve the system of equations, numerical methods such as direct methods (e.g., Gaussian elimination) or iterative methods (e.g., conjugate gradient method) can be employed. The choice of method depends on the size and structure of the system.

Once the system of equations is solved, the approximate solution of the PDE can be obtained by evaluating the basis functions at the nodes of the finite elements and combining them with the corresponding coefficients obtained from the solution of the system.

The accuracy of the numerical solution depends on various factors, such as the choice of basis functions, the size and shape of the finite elements, and the order of approximation. By refining the mesh and increasing the order of approximation, the accuracy of the solution can be improved.

Overall, the finite element method provides a powerful numerical technique for solving partial differential equations by approximating the solution within finite elements and constructing a system of algebraic equations. It is widely used in various fields, including engineering, physics, and computational mathematics.