Quasi-uniform finite-element mass inverses have summable graph decay

On a $d\ge 2$ reference simplex with barycentric coordinates ${\lambda }_0,\dots ,{\lambda }_d$, take the nodes $({\alpha }_0\mathrm{/}k,\dots ,{\alpha }_d\mathrm{/}k)$ for nonnegative integers ${\alpha }_j$ with ${\sum }_j{\alpha }_j=k$. They are unisolvent for ${\mathbb{P}}_k$. Indeed, use induction on $d+k$, with the zero-dimensional constant case and the degree-one vertex case as bases. For $k\ge 2$, restriction to the face ${\lambda }_0=0$ vanishes identically by the dimension induction, so the polynomial is ${\lambda }_{0}R$ with $\mathrm{deg}R\le k-1$. At nodes with ${\alpha }_0\ge 1$, the affine change ${\lambda }_0=(1+(k-1){\mu }_0)\mathrm{/}k$ and ${\lambda }_j=(k-1){\mu }_j\mathrm{/}k$ for $j\ge 1$ identifies the remaining nodes with the degree-$(k-1)$ barycentric lattice. The induction makes $R$ vanish at every node of that lattice, hence $R=0$. The number of barycentric nodes is $\left(\genfrac{}{}{0px}{}{k+d}{d}\right)$ by placing $d$ separators among $k$ stars; this equals the number of degree-at-most-$k$ monomials in $d$ variables, so it is $\dim {\mathbb{P}}_k$ and the unisolvent values define a basis. In $d=1$, the equally spaced $k+1$ nodes are unisolvent because a nonzero degree-$k$ polynomial has at most $k$ roots. On a shared face the two cells have the same face nodes; unisolvence of the face restriction makes their polynomial traces agree when the shared nodal values agree. The same face-node unisolvence shows that zero nodal coefficients on a boundary facet give its zero polynomial trace; in $d=1$ the boundary facets are the zero-valued endpoints. Unisolvence also means that local nodal interpolation reproduces every polynomial in ${\mathbb{P}}_k$, so the local basis sums to one and reproduces affine polynomials. Thus the constructed nodal functions are conforming and have the declared zero boundary values.

If ${\delta }_h(i,j)\le m$, a path of $m$ common-cell links gives $\mathrm{\mid }{z}_i-z_j\mathrm{\mid }\le mh$. Any cell containing such a node lies in $B(z_i,(m+1)h)$. In $d\ge 2$, quasi-uniformity and shape regularity give ${\rho }_K\ge \theta h\mathrm{/}\sigma$; each cell therefore contains an axis-parallel open cube of side $2\theta h\mathrm{/}(\sigma \sqrt{d})$ inside an inscribed ball. These cubes are disjoint and lie in the box $(z_i-(m+2)h,z_i+(m+2)h{)}^d$. By [F4], each has measure at least $(2\theta h\mathrm{/}(\sigma \sqrt{d}){)}^d$ and the outer box has measure $(2(m+2)h{)}^d$, so additivity and monotonicity bound the number of such cells by $C(m+1{)}^d$. In $d=1$, the intervals lie in an interval of length $2(m+1)h$ and have lengths at least $\theta h$, giving the same cell bound. Each cell contains a fixed number $N(d,k)$ of nodes, hence $\mathrm{\#}\{j:{\delta }_h(i,j)\le m\}\le C(d,k)(m+1{)}^d\mathrm{.}$ The same packing bound shows that the number of cells incident to any fixed node is uniformly bounded, since all such cells lie in $B(z_i,h)$.

On each reference cell, [F2] bounds the $L^r$ norm above and below by the ${\mathrm{\ell }}^r$ norm of the finitely many nodal coefficients; it also bounds the reference gradient norm above by that coefficient norm. Affine scaling then gives, for $1\le r<\mathrm{\infty }$, $\parallel {\sum }_{i\in I(K)}{c}_i{\varphi }_i{\parallel }_{L^r(K)}^r\asymp h_K^d{\sum }_{i\in I(K)}\mathrm{\mid }{c}_i{\mathrm{\mid }}^r,\parallel \mathrm{\nabla }{\sum }_{i\in I(K)}{c}_i{\varphi }_i{\parallel }_{L^r(K)}\le C{h}_K^{d\mathrm{/}r-1}\mathrm{\parallel }c{\mathrm{\parallel }}_{{\mathrm{\ell }}^r(I(K))}\mathrm{.}$ For $r=\mathrm{\infty }$ replace the first factor $h_K^{d\mathrm{/}r}$ by $1$. The same statements follow on an interval from $x=a+h_{K}t$ and the fixed one-dimensional nodal basis. Since $\theta h\le h_K\le h$, all cell scales are uniform in $h$.

For finite $r$, sum the local $L^r$ equivalence in step 2.1 over all cells. Each cell contains a fixed number $N(d,k)$ of nodes, and each global node is incident to uniformly many cells by the packing bound in step 1.2 (in $d=1$ at most two intervals meet at a node). Thus each global coefficient occurs between one and a uniformly bounded number of times, giving the asserted global lower and upper ${\mathrm{\ell }}^r$ scaling. Taking maxima of local polynomial norm equivalences gives the $r=\mathrm{\infty }$ scaling.

The interior-node mass matrix is symmetric positive definite because $c^{T}Mc=\mathrm{\parallel }{\sum }_i{c}_i{\varphi }_i{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2$, which is positive for every nonzero coefficient vector by the lower scaling in step 3.1. Taking $r=2$ there shows $a\mathrm{\parallel }c{\mathrm{\parallel }}_2^2\le c^T{A}_{h}c\le b\mathrm{\parallel }c{\mathrm{\parallel }}_2^2$ for positive constants independent of the mesh.

Put $\beta =2b$ and $B=I-A_h\mathrm{/}\beta$. By step 4.1 and [F3], $\mathrm{\parallel }B{\mathrm{\parallel }}_{2\to 2}\le q:=1-a\mathrm{/}(2b)<1$. The matrix $A_h$ has zero $ij$ entry unless $i=j$ or $i,j$ occur in a common cell. Consequently $(B^n{)}_{ij}=0$ for $n<{\delta }_h(i,j)$, since a nonzero product of $n$ entries would give a graph path of length at most $n$. The norm-convergent Neumann series gives $A_h^{-1}={\beta }^{-1}{\sum }_{n=0}^{\mathrm{\infty }}{B}^n,\mathrm{\mid }(A_h^{-1}{)}_{ij}\mathrm{\mid }\le {\beta }^{-1}{\sum }_{n\ge {\delta }_h(i,j)}{q}^n=\frac{q^{{\delta }_h(i,j)}}{\beta (1-q)}\mathrm{.}$ If ${\delta }_h(i,j)=\mathrm{\infty }$, every summand is zero and the inverse entry is zero.

Grouping the entry bound of step 5.1 by graph distance and applying the ball count of step 1.2 gives each absolute row sum at most $C{\sum }_{m\ge 0}(m+1{)}^d{q}^m<\mathrm{\infty }$; the series converges by the ratio test because its term ratio tends to $q<1$. Symmetry of $A_h^{-1}$ gives the same column bound. These constants do not depend on the number of cells.