Isogeometric analysis (IGA) is a numerical technique for partial differential equations, which was introduced in 2005 and received much attention since then. IGA was shown to be more accurate compared to the classical finite element analysis (FEA) for a given number of degrees of freedom. At the same time, the increased support of IGA basis functions compared to FEA ones strengthens the interconnection between the subdomains and leads to degradation of performance of linear solvers, thus increasing both time and memory requirements of IGA.
Recently, we proposed the optimally-blended schemes and their equivalent quadrature rules that yield two extra orders of convergence and improve the accuracy of Galerkin methods on uniform and non-uniform meshes for different polynomial orders and continuity of the basis. This reduction in approximation error comes at no additional computational cost. Moreover, the number of quadrature points can be minimized further leading to computationally less expensive quadrature rules.
I will also discuss the spectral properties of IGA with reductions of the continuity at certain hyperplanes in the mesh that allow for large savings in the computational cost, though producing artefacts in the spectra, such as stopping bands and outliers. A unified description of these effects will be given for both IGA and FEA.
In the second part of my talk, I will show our recent progress in parallel modelling and inversion for geophysical applications and present several examples of challenging geophysical scenarios where parallel computational resources are necessary for reducing the solution time to an acceptable time frame.