page_title: Finite element method
section_title: Discretization
hierarchical_section_title: Finite element method / Discretization
context_page_description: The finite element method is the most widely used method for solving problems of engineering and mathematical models. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
  The FEM is a particular numerical method for solving partial differential equations in two or three space variables. To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points.
  The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.
  The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.
context_section_description: P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:
  Find   such that
  with a finite-dimensional version:
  (3) Find  such that
  where  is a finite-dimensional subspace of . There are many possible choices for  (one possibility leads to the spectral method). However, for the finite element method we take  to be a space of piecewise polynomial functions.
caption: A function in 
    
      
        
          
            H
            
              0
            
            
              1
            
          
          ,
        
      
      {\displaystyle H_{0}^{1},}
    
   with zero values at the endpoints (blue), and a piecewise linear approximation (red)
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