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

Abstract

Abstract. We present two novel coarse spaces ( H 1 - and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>(</m:mo> <m:mo form="prefix">curl</m:mo> <m:mo>)</m:mo> </m:mrow> </m:math> $H(\operatorname{curl})$ -conforming) based on element agglomeration on unstructured tetrahedral meshes. Each H 1 -conforming coarse basis function is continuous and piecewise-linear with respect to an original tetrahedral mesh. The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>(</m:mo> <m:mo form="prefix">curl</m:mo> <m:mo>)</m:mo> </m:mrow> </m:math> $H(\operatorname{curl})$ -conforming coarse space is a subspace of the lowest order Nédélec space of the first type. The H 1 -conforming coarse space exactly interpolates affine functions on each agglomerate. The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>(</m:mo> <m:mo form="prefix">curl</m:mo> <m:mo>)</m:mo> </m:mrow> </m:math> $H(\operatorname{curl})$ -conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>(</m:mo> <m:mo form="prefix">div</m:mo> <m:mo>)</m:mo> </m:mrow> </m:math> $H(\operatorname{div})$ - and L 2 -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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo form="prefix">curl</m:mo> </m:math> $\operatorname{curl}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo form="prefix">div</m:mo> </m:math> $\operatorname{div}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo form="prefix">grad</m:mo> </m:math> $\operatorname{grad}$ differential operators.