The value of continuity: Refined isogeometric analysis and fast direct solvers
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We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce $C^0$-separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method ``refined Isogeometric Analysis (rIGA)". To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between $p^2$ and $p^3$, with $p$ being the polynomial order of the discretization. Numerical results indicate that our proposed rIGA method delivers a speed-up factor proportional to $p^2$. In a $2D$ mesh with four million elements and $p = 5$, the linear system resulting from rIGA is solved $22$ times faster than the one from highly continuous IGA. In a $3D$ mesh with one million elements and $p = 3$, the linear rIGA system is solved $15$ times faster than the IGA one.