Light PhD Seminar: Refined Isogeometric Analysis (rIGA): A multi-field application

Abstract: 
Refined Isogeometric Analysis (rIGA) is a discretization method used to solve numerical problems governed by partial differential equations (PDEs). Starting from a highly continuous Isogeometric Analysis (IGA) discretization, rIGA reduces the continuity over certain hyperplanes that split the mesh into subdomains. This method maximizes the performance of direct solvers by reducing the continuity until $C^0$ over selected hyperplanes, which act as separators during the elimination of the degrees of freedom (DoF). By doing so, the solution time and best approximation error are simultaneously improved. In particular, rIGA delivers a speedup factor with respect to IGA that is proportional to $p^2$ when solving Laplace based problems in 2D and 3D, with $p$ being the polynomial degree.

In this talk, we introduce rIGA method to solve multi-field problems. We consider an incompressible fluid flow problem on bounded domains that includes the pressure and the vectorial velocity of the fluid. We use a spline-based generalization of the Raviart-Thomas finite element spaces to approximate the velocity field.

We show that rIGA delivers a reduction in the computational cost when solving incompressible fluid flow problems that asymptotically reaches $mathcal{O}(p^2)$, and it provides better accuracy than $C^{p-1}$ IGA. For multi-field problems, however, we require larger problems to arrive at the asymptotic limit and reach the maximum possible savings since the system involves more equations. In our numerical 2D results, we observe a reduction factor in the computational cost of up to $p^2$. In 3D, the maximum reproducible problems are in the pre-asymptotic regime, and the maximum observed gain factors are of $mathcal{O}(p)$.