Algebraic Multigrid for High-Order Hierarchical $H(curl)$ Finite Elements

Abstract

Classic multigrid methods are often not directly applicable to nonelliptic problems such as curl-type partial differential equations (PDEs). Curl-curl PDEs require specialized smoothers that are compatible with the gradient-like (near) null space. Moreover, recent developments have focused on replicating the grad-curl-div de Rham complex in a multilevel hierarchy through smoothed aggregation based algebraic multigrid. These approaches have been successful for Nédélec finite elements (i.e., H(curl) edge elements), but do not extend naturally to high-order representations. In this paper we consider hierarchical high-order Whitney elements for the curl-curl eddy current problem and devise a scalable multilevel approach. Our method generates a hierarchy similar to p-type multigrid, which results in a coarse level that is amenable to further coarsening through the established process of a multilevel complex. The natural hierarchy of the elements induces an effective interpolation operator and motivates the construction of a compatible gradient smoothing process. We detail the multilevel solver for a hierarchical H(curl) basis and present numerical results in support of the method.

Algebraic multigrid methods

“…As indicated by the convergence estimates, for a given smoother, the desirable coarse spaces should capture or well approximate the lower end of the spectrum of the relaxed matrix or . This is usually referred to as near-null space (Treister and Yavneh 2015, Lai and Olson 2011, Xu 2009, Brezina et al. 2006).…”

Algebraic Multigrid Methods

“…As indicated by the convergence estimates, for a given a smoother, the desirable coarse spaces should capture or approximate well the lower end of the spectrum of the relaxed matrix RA or D −1 A. This is usually referred to as nearnull space (Treister and Yavneh 2015, Lai and Olson 2011, Xu 2009, (Brezina, Falgout, MacLachlan, Manteuffel, McCormick and Ruge 2006a).…”

An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell’s equations

“…int edgesMap [13] = {-1,-1,0,1,2,-1,3,-1,4,-1,-1,-1,5}; // static const int nvedge [6][2] = {{0,1},{0,2},{0,3},{1,2},{1,3},{2,3}}; int p20 [20]; for(int i=0; i<6; ++i) // edge dofs { int ii0 = Element::nvedge Then, we will save the linear combinations of the w , with coefficients given by the j-th column of V −1 (see Example 1), in the final basis functions wP20[j] , thus in duality with the chosen dofs: [15]; wtilde[p20 [12]] = +8*w[12]-4*w [13]; wtilde[p20 [13]] = -4*w [12]+8*w [13]; wtilde[p20 [14]] = +8*w[14]-4*w [15]; wtilde[p20 [15]] = -4*w [14]+8*w [15]; wtilde[p20 [16]] = +8*w[16]-4*w [17]; wtilde[p20 [17]] = -4*w [16]+8*w [17]; wtilde[p20 [18]] = +8*w[18]-4*w [19]; wtilde[p20 [19]] = -4*w [18]+8*w [19];…”

“…For Maxwell's equations in the time domain, for which an implicit time discretization yields at each step a positive definite problem, there are many good solvers and preconditioners in the literature: multigrid or auxiliary space methods, see e.g. [12,13,14,15] for low order finite elements, [16] for high order ones, and Schwarz domain decomposition methods, see e.g. [17,18].…”

“…wtilde[p20[0]] = +4*w[0]-2*w[1]-4*w[16]+2*w[17]-4*w[18]+2*w[19]; wtilde[p20[1]] = -2*w[0]+4*w[1]-2*w[16]-2*w[17]-2*w[18]-2*w[19]; wtilde[p20[2]] = +4*w[2]-2*w[3]-4*w[14]+2*w[15]+2*w[18]-4*w[19]; wtilde[p20[3]] = -2*w[2]+4*w[3]-2*w[14]-2*w[15]-2*w[18]-2*w[19]; wtilde[p20[4]] = +4*w[4]-2*w[5]+2*w[14]-4*w[15]+2*w[16]-4*w[17]; wtilde[p20[5]] = -2*w[4]+4*w[5]-2*w[14]-2*w[15]-2*w[16]-2*w[17]; wtilde[p20[6]] = +4*w[6]-2*w[7]-4*w[12]+2*w[13]+2*w[18]-4*w[19]; wtilde[p20[7]] = -2*w[6]+4*w[7]-2*w[12]-2*w[13]+4*w[18]-2*w[19]; wtilde[p20[8]] = +4*w[8]-2*w[9]+2*w[12]-4*w[13]+2*w[16]-4*w[17]; wtilde[p20[9]] = -2*w[8]+4*w[9]-2*w[12]-2*w[13]+4*w[16]-2*w[17]; wtilde[p20[10]] = +4*w[10]-2*w[11]+2*w[12]-4*w[13]+2*w[14]-4*w[15]; wtilde[p20[11]] = -2*w[10]+4*w[11]+4*w[12]-2*w[13]+4*w[14]-2*w…”