Substructuring, FETI-DP and BDDC — Examples

Multiplying the displayed matrix from [F1] by $\mathbf{1}$ gives zero, so the nonzero vector $\mathbf{1}$ is in the local nullspace.

Since the dual subproblem imposes no primal value or functional that changes on constants, $\mathbf{1}$ also satisfies every homogeneous local dual constraint. Thus its restricted quadratic form is singular, contradicting unique solvability if one attempted the local solve.

If one corner value were retained as primal, a dual correction would have to vanish there; a constant vector with that value zero is the zero vector, so this particular null mode would be removed as in [F3]. The counterexample therefore concerns a whole floating component with no anchor, not omission of one among several anchors.

On triangle $(a,b,c)$ the affine basis gradients are $(-1,0),(1,-1),(0,1)$ and its area is $1\mathrm{/}2$; on triangle $(a,c,d)$ they are $(0,-1),(1,0),(-1,1)$ and its area is $1\mathrm{/}2$. Adding the two local Gram matrices of these gradients gives the displayed $K_i$.

The displayed matrix has zero row sums, and its quadratic form is $\frac{1}{2}[(u_a-u_b{)}^2+(u_b-u_c{)}^2+(u_c-u_d{)}^2+(u_d-u_a{)}^2]$. It is therefore nonnegative with kernel exactly the constant vector, agreeing with [F2].

For a partially assembled trace of zero energy, both nonnegative local energies vanish. By step 2.1 each local vector is constant; its imposed zero top-side values make that constant zero. Hence the partially assembled quadratic form has trivial kernel and is positive definite, directly, without needing the all-vertices-primal criterion in [F3]. The only shared nodes are the primal endpoints, so their equalities enforce full interface continuity. Theorem [F4] then identifies the condensed interface solution and the load-dependent interior reconstruction, whose homogeneous trace-dependent part is harmonic, with the original global finite-element solution.

The determinant of $A$ is $3$, so $A^{-1}=\frac{1}{3}\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$. The BDDC coarse operator is $R^{T}AR=2$, while $E_0{A}^{-1}{E}_0^T=2\mathrm{/}3$; their product is $4\mathrm{/}3$.

The dual operator is $B{A}^{-1}{B}^T=2\mathrm{/}3$, and its preconditioner is $B_{D}A{B}_D^T=2$; their product is also $4\mathrm{/}3$. Direct multiplication gives the three identities in the Example, so the abstract assumptions hold.

Both operators are one-dimensional, hence each has the single eigenvalue computed in steps 1.1 and 1.2. The common value is positive and different from $1$, consistent with [F3].

A dense Cholesky factorization of an $m\times m$ SPD matrix uses a constant multiple of $m^3$ arithmetic operations and stores $O(m^2)$ entries. Repeating on $N$ local matrices gives total $O(N{m}^3)$ work, and perfect concurrency reduces the local phase to $O(m^3)$ time.

Since $n_c$ is proportional to $N$, dense coarse Cholesky uses $O(N^3)$ arithmetic and $O(N^2)$ storage. If $H\mathrm{/}h$ stays fixed while the number of subdomains grows, [F1] leaves the iteration condition bound unchanged, but the coarse direct-solve work grows cubically.