System matrix assembly for isogeometric (i.e., spline-based) discretizations of partial differential equations is more challenging than for classical finite elements, due to the increased polynomial degrees and the larger (and hence more overlapping) supports of the basis functions. The global tensor-product structure of the discrete spaces employed in isogeometric analysis can be exploited to accelerate the computations, using sum factorization, precomputed look-up tables, and tensor decomposition. We generalize the third approach by considering partial tensor decompositions. We show that the resulting new method preserves the global discretization error and that its computational complexity compares favorably to the existing approaches. Moreover, the numerical realization simplifies considerably since it relies on standard techniques from numerical linear algebra.
In this paper we present a space-time isogeometric analysis scheme for the discretization of
parabolic evolution equations with diffusion coefficients depending on both time and space variables.
The problem is considered in a space-time cylinder in {\mathbb{R}^{d+1}}, with {d=2,3}, and is discretized using higher-order and highly-smooth spline spaces. This makes the matrix formation task very challenging from a computational point of view. We overcome this problem by introducing a low-rank decoupling of the operator into space and time components. Numerical experiments demonstrate the efficiency of this approach.
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