Variatsioonilõikemeteodite

Variatsioonilõikemeteodite are a family of numerical methods used to approximate solutions to differential equations. The core idea is to transform a differential equation into an integral equation, which is then solved by minimizing a functional. This functional is derived from the original differential equation through a process called variational formulation or by applying integral formulations. The domain of the problem is then discretized into a finite number of elements. Within each element, the unknown solution is approximated using simple basis functions, such as polynomials. These basis functions are defined over each element and are typically continuous across element boundaries.

The method involves substituting these approximate basis functions into the integral formulation of the problem. This process results in a system of algebraic equations. The unknowns in this system are the coefficients of the basis functions, which represent the approximate values of the solution at specific points called nodes. These nodes are located at the boundaries of the elements. Solving this system of algebraic equations yields the approximate solution to the original differential equation. Variatsioonilõikemeteodite are widely applied in various fields of engineering and physics, including structural analysis, fluid dynamics, and heat transfer, due to their ability to handle complex geometries and boundary conditions.