Quadrature representation of finite element variational forms

This chapter addresses the conventional run-time quadrature approach for the numerical integration of local element tensors associated with finite element variational forms, and in particular automated optimizations that can be performed to reduce the number of floating point operations. An alternative to the run-time quadrature approach is the tensor representation presented in Chapter 8. Both the quadrature and tensor approaches are implemented in FFC (see Chapter 11). In this chapter we discuss four strategies for optimizing the quadrature representation for run-time performance of the generated code and show that optimization strategies lead to a dramatic improvement in run-time performance over a naive implementation. We also examine performance aspects of the quadrature and tensor approaches for different equations, and this will motivate the desirability of being able to choose between the two representations