Numerical Analysis Questions Medium
The finite element method (FEM) is a numerical technique used to solve partial differential equations (PDEs) by dividing the problem domain into smaller subdomains called finite elements. These finite elements are interconnected at specific points called nodes, forming a mesh or grid.
To solve a PDE using the FEM, the first step is to discretize the problem domain into finite elements. This is typically done by subdividing the domain into triangles or quadrilaterals in 2D problems, or tetrahedra or hexahedra in 3D problems. Each finite element is defined by a set of nodes and has a specific shape function associated with it.
Next, the governing PDE is approximated by a set of algebraic equations using the principle of weighted residuals. This involves multiplying the PDE by a set of weight functions, which are typically chosen to be piecewise continuous functions defined over each finite element. The resulting weighted residual equations are then integrated over each finite element.
The integration process involves evaluating the integrals numerically using techniques such as Gaussian quadrature. This allows the PDE to be represented by a system of algebraic equations, known as the finite element equations. These equations relate the unknown values of the solution at the nodes of the finite elements.
The next step is to solve the system of finite element equations to obtain the solution. This is typically done by assembling the finite element equations into a global system of equations, which can be solved using techniques such as direct methods (e.g., Gaussian elimination) or iterative methods (e.g., conjugate gradient method). The solution obtained represents an approximation to the true solution of the PDE.
Finally, the solution is post-processed to obtain the desired quantities of interest. This may involve evaluating the solution at specific points, calculating derivatives, or computing integrals over certain regions of the domain.
Overall, the finite element method provides a flexible and powerful approach for solving partial differential equations. It allows for the accurate and efficient numerical approximation of complex problems, making it a widely used technique in various fields such as engineering, physics, and applied mathematics.