stylistic:style_transfer, linguistic:translation
Choosing a basis

To complete the discretization, we must select a basis of V {\displaystyle V}. In the one-dimensional case, for each control point x k {\displaystyle x_{k}} we will choose the piecewise linear function v k {\displaystyle v_{k}} in V {\displaystyle V} whose value is 1 {\displaystyle 1} at x k {\displaystyle x_{k}} and zero at every x j, j ≠ k {\displaystyle x_{j},\;j\neq k}, i.e., v k (x) = { x − x k − 1 x k − x k − 1 if x ∈, x k + 1 − x x k + 1 − x k if x ∈, 0 otherwise, {\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in,\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in,\\0&{\text{ otherwise}},\end{cases}}}

for k = 1, …, n {\displaystyle k=1,\dots,n}; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function v k {\displaystyle v_{k}} per vertex x k {\displaystyle x_{k}} of the triangulation of the planar region Ω {\displaystyle \Omega }. The function v k {\displaystyle v_{k}} is the unique function of V {\displaystyle V} whose value is 1 {\displaystyle 1} at x k {\displaystyle x_{k}} and zero at every x j, j ≠ k {\displaystyle x_{j},\;j\neq k}.

Depending on the author, the word "element" in the "finite element method" refers to the domain's triangles, 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" with "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method is not restricted to triangles (tetrahedra in 3-d or higher-order simplexes in multidimensional spaces). Still, it 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 an approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques; the most popular are: