Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods
Provides optimal preconditioning for adaptive IGA boundary element methods, addressing a key computational bottleneck in numerical simulations for engineering and physics.
The paper develops multilevel diagonal preconditioners for isogeometric boundary elements on locally refined 2D meshes, proving mesh-size and refinement-level independent condition numbers, thus achieving optimal computational complexity for hypersingular and weakly-singular integral equations.
We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.