arXiv:2610.31982v1 Announce Type: new Abstract: Efficient matrix-free preconditioning remains a challenge for discontinuous Galerkin spectral element discretisations of elliptic problems, particularly for general element shapes with collapsed-coordinate bases, such as prisms, tetrahedra and pyramids. Multigrid provides a promising framework, but its efficiency depends strongly on the effectiveness and computational cost of the smoother. For modal bases, point Jacobi can provide insufficient smoothing, whereas applying block Jacobi through stored dense inverses incurs substantial storage and computational costs. We propose a matrix-free approximate elemental inverse, an approximation to the inverse of each elemental diagonal block, constructed from inverse modal-to-nodal transforms and an element-dependent sparse matrix. Its application exploits generalised sum-factorisation and partial sum-factorisation in collapsed coordinates, avoiding the explicit storage and multiplication of large dense matrices. The method is evaluated as both a standalone preconditioner and a multigrid smoother on hexahedral, prismatic, tetrahedral and pyramidal elements. Although the proposed approximation generally increases iteration counts compared to exact block Jacobi, the cost per iteration is far lower. This trade-off yields substantial reductions in total solution time for hexahedral and prismatic elements, while achieving comparable performance for tetrahedral and pyramidal elements. This study can also serve as a practical guide for tuning the convergence of the discontinuous Galerkin interior-penalty Poisson solver.
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