Add some documentation on basis functions

Author: nmnobre

mkdocs.yml

@@ -36,6 +36,7 @@ nav:
   - 'HowTo Articles': howto/howto-index.md
   - 'Finite Elements': fem.md
   - '_Primal and Dual Vectors': pri-dual-vec.md
+  - '_Finite Element Basis Functions': basis-functions.md
   - '_Bilinear Form Integrators': bilininteg.md
   - '_Linear Form Integrators': lininteg.md
   - '_Integration': integration.md

src/basis-functions.md

@@ -0,0 +1,114 @@
+tag-fem:
+
+# Basis Functions
+
+The finite element method is based on the notion that a smooth function can be
+approximated by a linear combination of piece-wise smooth functions (typically
+piece-wise polynomials) called *basis functions*. The coefficients of the
+linear combination are called *degrees of freedom (DoFs)* and are linear
+functionals of the approximated function, e.g. values at interpolation points
+or integrals over submanifolds of the domain. No matter the space the problem
+is formulated in, H1, H(curl), H(div) or L2, or more precisely the discrete
+space the problem is [weakly-formulated](fem_weak_form.md) in, there is a wide
+range of possible basis function choices spanning the very same discrete space.
+
+## One-Dimensional Polynomial Basis Functions
+
+Consider the $[0,1]$ interval on the real line. We distinguish between open and
+closed function bases. An *open* basis does *not* interpolate the interval
+endpoints, whereas a *closed* basis does. If a basis is entirely interpolatory,
+in the sense that all its associated DoFs correspond to values of the
+approximated function, we call it a *nodal* basis.
+
+<center>**Third order, one-dimensional basis functions on [0,1]**</center>
+![](img/basis-functions.pdf)
+
+All of the following are bases of the space of univariate polynomials of degree
+less than or equal to $n$:
+
+- `GaussLegendre`: open and nodal basis; the basis of order $n$ interpolates
+all the quadrature points of the Gauss-Legendre quadrature rule with $n+1$
+points. These points are the zeros of the Legendre polynomial of degree $n+1$.
+
+- `OpenUniform`: open and nodal basis; the basis of order $n$ interpolates all
+the quadrature points of an open Newton-Cotes quadrature rule with $n+1$
+equidistant points. These points partition the $[0,1]$ interval in $n+2$
+subintervals of equal length, $x_{i \in {0,...,n}} = (i+1)/(n+2)$.
+
+- `OpenHalfUniform`: open and nodal basis; the basis of order $n$ interpolates
+all the quadrature points of an open Newton-Cotes quadrature rule with $n+1$
+equidistant points. These are the midpoints of $n+1$ subintervals of equal
+length partitioning the $[0,1]$ interval,
+$x_{i \in {0,...,n}} = (1/2+i)/(n+1)$.
+
+- `IntegratedGLL`: open basis; the basis of order $n$ histopolates all the
+subintervals in-between the quadrature points of the Gauss-Lobatto quadrature
+rule with $n+2$ points. These points are the $[0,1]$ interval endpoints plus
+the zeros of the first derivative of the Legendre polynomial of degree $n+1$.
+
+- `GaussLobatto`: closed and nodal basis; the basis of order $n$ interpolates
+all the quadrature points of the Gauss-Lobatto quadrature rule with $n+1$
+points. These points are the $[0,1]$ interval endpoints plus the zeros of the
+first derivative of the Legendre polynomial of degree $n$.
+
+- `ClosedUniform`: closed and nodal basis; the basis of order $n$ interpolates
+all the quadrature points of the closed Newton-Cotes quadrature rule with $n+1$
+equidistant points. These points partition the $[0,1]$ interval in $n$
+subintervals of equal length, $x_{i \in {0,...,n}} = i/n$.
+
+- `ClosedGL`: closed and nodal basis; the basis of order $n$ interpolates the
+endpoints plus the midpoints of the subintervals in-between all the quadrature
+points of the Gauss-Legendre quadrature rule with $n$ points.
+
+- `Positive`: closed basis; the basis of order $n$ is simply the Bernstein
+polynomial basis of degree $n$ and only interpolates the $[0,1]$ interval
+endpoints, $b_{i \in {0,...,n}}(x) = {n \choose i} x^{i}(1-x)^{n-i}$.
+
+## Finite Element Collections and Elements of Different Types
+
+When constructing a [finite element collection](FiniteElementCollection) for
+a given order and dimension, the user can often specify one, for
+[H1](H1_FECollection) and [L2](L2_FECollection) spaces, or two, for
+[ND](ND_FECollection) and [RT](RT_FECollection) spaces, of the one-dimensional
+bases described above. In most cases the user can simply use the provided
+defaults, Gauss-Lobatto for H1 spaces, Gauss-Lobatto plus Gauss-Legendre for
+ND and RT spaces, and Gauss-Lobatto for L2 spaces.
+
+For tensor product elements, e.g. quadrilaterals or hexahedra, the basis
+functions are just products of one-dimensional closed and/or open basis
+functions. For simplices, e.g. triangles or tetrahedra, the process of
+constructing the basis functions is slightly more involved, but can essentially
+be pictured as starting from the one-dimensional bases above and extending in a
+triangular lattice on the faces and in the interior, thus retaining their
+property as open, closed and/or nodal. In practice, in MFEM, the basis
+functions for simplices are obtained as products of a hierarchical,
+one-dimensional basis of Chebyshev polynomials of the first kind. The user can
+use the [`display-basis` miniapp](tools.md#display-basis) to visualize various
+types of basis functions on a single mesh element of their choice.
+
+## Some Notes on Choosing Basis Functions
+
+Occasionally, the user might find the default basis functions are not suitable
+for a given problem formulation. The following is a non-exhaustive list of
+considerations to take into account when choosing what basis functions to use.
+
+First, due to their nature, not all collections accept all function bases. For
+instance, H1 spaces are continuous and thus require a basis which is
+interpolatory at the element boundaries, i.e. a closed basis. On the contrary,
+L2 spaces do not necessarily have DoFs at the element boundaries and can be
+spanned by an open basis, or any basis in fact. If an open basis is selected
+though, [extra care](fem_bc.md#discontinuous-galerkin-formulations) is needed
+in order to specify Dirichlet boundary conditions.
+
+Second, nodal basis functions are particularly efficient to evaluate using
+barycentric interpolation, and to integrate, specially if the corresponding
+collocated quadrature rule is used.
+
+Third, Low-Order-Refined (LOR) discretizations are only spectrally equivalent
+to the original high-order system if using a Gauss-Lobatto basis for H1 spaces
+and Gauss-Lobatto plus IntegratedGLL bases for both ND and RT spaces. Note that
+if a different choice is made, MFEM issues a warning but will still proceed.
+
+<script type="text/x-mathjax-config">MathJax.Hub.Config({TeX: {equationNumbers: {autoNumber: "all"}}, tex2jax: {inlineMath: [['$','$']]}});</script>
+<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_HTML"></script>
+

src/fem.md

@@ -84,6 +84,12 @@ Finite element approximations are almost always less smooth than the
 solutions we hope to approximate.  Weak formulations provide a means
 of approximating derivatives of non-differentiable functions.
 
+### [Basis Functions](basis-functions.md)
+
+Basis functions are the building blocks of finite element approximations.
+This page gives an overview of the one-dimensional polynomial basis functions
+that are used to build the basis functions spanning higher-dimensional spaces.
+
 ### [Boundary Conditions](fem_bc.md)
 
 The types of available boundary conditions and how to apply them