The Construction of the Coarse de Rham Complexes with Improved Approximation Properties

Abstract.

We present two novel coarse spaces (H¹- and H(curl)-conforming) based on element agglomeration on unstructured tetrahedral meshes. Each H¹-conforming coarse basis function is continuous and piecewise-linear with respect to an original tetrahedral mesh. The H(curl)-conforming coarse space is a subspace of the lowest order Nédélec space of the first type. The H¹-conforming coarse space exactly interpolates affine functions on each agglomerate. The H(curl)-conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the H(div)- and L₂-conforming spaces developed previously in [Numer. Linear Algebra Appl. 19 (2012), 414–426], the newly constructed coarse spaces form a sequence (with respect to exterior derivatives) which is exact as long as the underlying sequence of fine-grid spaces is exact. The constructed coarse spaces inherit the approximation and stability properties of the underlying fine-grid spaces supported by our numerical experiments. The new coarse spaces, in addition to multigrid, can be used for upscaling of broad range of PDEs involving curl, div and grad differential operators.