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

Abstract

Abstract. We present two novel coarse spaces (H1- and $H(\operatorname{curl})$-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 $H(\operatorname{curl})$-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 $H(\operatorname{curl})$-conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the $H(\operatorname{div})$- 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 $\operatorname{curl}$, $\operatorname{div}$ and $\operatorname{grad}$ differential operators.