Solution Costs of Isogeometric Analysis for ORNL's Computational and Applied Mathematics Group

I will present an analysis of the cost of multi-frontal direct and iterative solvers when applied to linear systems resulting from a discretized elliptic partial differential equation with the higher-order continuous B-spline basis, popularly known as isogeometric analysis.

Since 2005, the engineering research community has invested heavily in the development of isogeometric analysis, originally due to a desire to more tightly couple analysis to geometry representations, but additionally because of many favorable qualities of the B-spline basis. Among these is an observable improvement in the constant of the error convergence when using the spaces to discretize partial differential equations. This improvement grows as the polynomial order of the space grows, reaching as large as one to two orders of magnitude relative to the traditional finite element spaces. This sparked great interest in using the B-spline basis as an economic high polynomial order method.

However, this assessment ignored the differences in the costs of solving the resulting linear systems. We found that relative to the traditional C^0 finite element spaces, the maximally continuous C^(p-1) basis, where p is the polynomial order of the space, is O(p^3) times more expensive to solve when using a multi-frontal direct solver and O(p^2) when using iterative solvers.

In this talk, I will discuss the derivation of these estimates and present results which support the conclusions. This talk will be of value to those interested in using isogeometric analysis in practice or in understanding how a multi-frontal direct solver works. More broadly, this talk advocates for a more holistic view of developing numerical methods which includes solution costs.