A sparse-grid isogeometric solver
For researchers in computational PDEs, this work provides a practical method to enhance IGA solver performance, but the results are largely confirmatory of known sparse-grid benefits.
The paper investigates the application of sparse-grid construction (combination technique) to isogeometric analysis (IGA) solvers, finding that it improves efficiency for smooth solutions and can be beneficial for non-smooth solutions with a-priori knowledge of singularities, while also enabling straightforward parallelization of serial IGA solvers.
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.