Finite Element Methods - Discretization - Choosing A Basis

Choosing A Basis

To complete the discretization, we must select a basis of . In the one-dimensional case, for each control point we will choose the piecewise linear function in whose value is at and zero at every, i.e.,

for ; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function per vertex of the triangulation of the planar region . The function is the unique function of whose value is at and zero at every .

Depending on the author, the word "element" in "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". Finite element method is not restricted to triangles (or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle).

Examples of methods that use higher degree piecewise polynomial basis functions are the hp-FEM and spectral FEM.

More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the 'exact' solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:

• moving nodes (r-adaptivity)
• refining (and unrefining) elements (h-adaptivity)
• changing order of base functions (p-adaptivity)
• combinations of the above (hp-adaptivity).