Tensor product structures for polynomials and quadrature

Rationale

Most of the local objects in dealii have a tensor product structure. Quadrature formulas are tensor products of one-dimensional formulas. Shape functions are tensor products of polynomials in different coordinate directions. Even spaces like Nédéléc and Raviart-Thomas have such structures in each component. Katharina Kormann and Martin Kronbichler have shown that exploiting these structures can boost performance of local integrals considerably. Therefore, adapting finite element and quadrature classes to their concepts is a valuable task.

An important ingredient required for the optimization of local integrals is that the length of all loops is known at compile time. Therefore, the number of quadrature points and the number of shape functions must become template arguments.

The class FEValues was originally introduced to allow for optimizations based on the internal structure of finite element shape function spaces, but this was never exploited. Additionally, FEValues stores a lot of data to avoid recomputation. Data, which is highly redundant. At a time where computers had a single working thread and computations were expensive, this was in part justified. Now, that most algorithms become memory-bound, we must consider retiring FEValues.

Discussion

• Implementation of hp methods
• Efficient implementation of methods where we do not have tensor product structure
  • XFEM
  • Overlapping nonmatching grids
• Note: hp adaptivity is supported by the FEEvaluation class.

Design and limitations of the FEEvaluation class

The matrix-free implementation uses the class FEEvaluation instead of FEValues. Different from the FEValues class, information about the degrees of freedom and the mapping from physical to unit cell is precomputed and stored. The degrees of freedom are numbered MPI-local and the constraints are incorporated into the DoF list in order to provide direct array access into vectors. This enables e.g. fast evaluation of the shape functions at the quadrature points. On the other hand, only values, gradients, etc. that are precomputed can be accessed while the FEValues object enables on-the-fly computations of more general evaluations. The main task would therefore be to merge the different functionalities of the two classes.

Tensor product quadrature

It is suggested to introduce a new class template inheriting from Quadrature, say TensorProductQuadrature. In a compatibility mode, this class fills the fields in Quadrature (deferred execution preferred). Quadrature points are stored in a TensorProductPoints, where the access operator [] returns a Point<dim> created on the fly. Additionally, it allows direct access to the one-dimensional point sets. Similarly, we can consider a class TensorProductWeights.

Note that while we are currently not making use of this (this is false: there are classes to integrate singularities which are not based on tensor product quadrature formulas...), Quadrature is not restricted to tensor products. Therefore, TensorProductQuadrature is not going to be an equivalent replacement.

Discussion

• We will need anisotropic tensor products. Do we want an isotropic class as well?
• If anisotropic, are variadic templates the only option?
• The order of the quadrature point (do we make any assumptions yet?)
  • Currently: lexicographic, low to high abscissa
  • Low to high weight for numerical stability?
  • End points of intervals first to conform to finite elements?
• Not inheriting from Quadrature would on one hand mean that we have two independent quadrature sets, on the other hand we would not need to link to the library, which is important for off-loading.

Draft of the classes

The convention for the indices shall be that d=1,...,dim, di enumerates the coordinates in direction d, and i enumerates all quadrature points.

template <typename number, int dim, int...>
class TensorProductPoints
{
  public:
    template <typename number2>
    void set_coordinates (const unsigned int d, const std::vector<number2>&);
    
    number operator() (const unsigned int d, const unsigned int di) const;
    Point<dim> point(const unsigned int i) const;
};

The class TensorProductWeights is similar and TensorProductQuadrature combines the two.

Tensor product structure of polynomials

As Katharina and Martin point out, combining tensor product quadrature with tensor product polynomials can generate very efficient code. This has been known to the spectral element community for a while, but widely ignored by us. It is implemented in the matrix free framework, but only for certain elements and with limited functionality. One goal of these new structures is making efficient matrix free computations available throughout the library.

Example: Lagrange polynomials

Given polynomials of degree p-1 and q quadrature points in each direction, the current FEValues in 3D computes and stores p^3 q^3 shape function values, three times as many derivatives and nine times as many second derivatives. In (isotropic) tensor product representation, we need p q values, derivatives, and second derivatives. This means, that now even higher derivatives and 4D become feasible. As for computation of these values, some polynomial spaces allow for a recursive computation of the values of higher order members. Thus, here is another point where we can reduce the computational complexity by a factor up to p.

Example: vector valued elements

FE_Nedelec as well as FE_RaviartThomas can be implemented such that each component is an anisotropic tensor product.

Example: FE_DGP

This example differs from the previous ones in that all multivariate polynomials are still products of 1D ones, but that not all possible combinations are admitted. This is true for serendipity elements as well. For these we need truncated tensor products.

Discussion

• The matrix free code currently does a renumbering to tensor product numbering of the shape functions, which is the inversion of a similar process when we generate the functions. Should we reconsider the ordering in DoFHandler?
• FEEvaluation currently knows the elements FE_Q, FE_DGQ, FE_DGP and FE_Q_DG0.