For a bounded bilinear form \(a ( \cdot ,\cdot ): X\times X \rightarrow \mathbb {R}\) where \(X\) is a Hilbert space, it has an associated bounded linear operator \(A: X \rightarrow X'\) which satisfies \(a(u,v)=\langle Au,v \rangle \) for all \(u\) and \(v\) in \(X\). In other words, the behavior of the bilinear form \(a(\cdot ,\cdot )\) as a binary function (usually the operand is also a function) can be fully determined by the linear operator \(A\) as a unary function plus a definition of the duality pairing 1. Because of the boundedness of \(a(\cdot ,\cdot )\), there exists a positive constant \(c_2^A\) such that \begin{equation} a ( u,u ) = \langle Au,u \rangle \leq c_2^A \lVert u \rVert _X^2 \quad \forall u\in X. \end{equation} Meanwhile, if \(A\) is \(X\)-elliptic, there exists a positive constant \(c_1^A\) such that \begin{equation} \langle Au,u \rangle \geq c_1^A \lVert u \rVert _X^2 \quad \forall u\in X. \end{equation} We can consider \(a(u,u)\) or \(\langle Au,u \rangle \) as an energy norm for \(u\) (Brenner, page 5) (see also Understanding about energy norm used in Galerkin method), while \(a(u,u)/\lVert u \rVert _X^2\) or \(\langle Au,u \rangle /\lVert u \rVert _X^2\) as a normalized energy norm. Therefore, \([c_1^A,c_2^A]\) is the range of this normalized energy norm, which is also the range of the spectrum of the operator \(A\).

If we consider a finite dimensional subspace \(V_h\) of \(V\), the corresponding function for \(u\) becomes \(u_h\). Let \(\{ \varphi _k \}_{k=1}^N\) be the basis of \(V_h\), then the function \(u_h\) has this expansion \begin{equation} u_h=\sum _{k=1}^N u_k\varphi _k. \end{equation} and can be discretized into a vector \(\underline {u}\in \mathbb {R}^N\). The bilinear form associated with \(A\) can be discretized into a matrix \(\mathcal {A}\), whose coefficient is \begin{equation} \mathcal {A}_{ij}=\langle A\varphi _j,\varphi _i \rangle . \end{equation} \(\mathcal {A}\) has similar boundedness and ellipticity properties as the operator \(A\): \begin{equation} \tilde {c}_1^A \lVert \underline {u} \rVert ^{2} \leq \underline {u}^{\mathrm {T}}\mathcal {A}\underline {u} \leq \tilde {c}_2^A \lVert \underline {u} \rVert ^{2} \quad \forall \underline {u}\in \mathbb {R}^N. \end{equation} According to Understanding about ellipticity of operators, \(\tilde {c}_1^A\) and \(\tilde {c}_2^A\) correspond to the minimum and maximum eigenvalues of \(\mathcal {A}\) and the range \([\tilde {c}_1^A,\tilde {c}_2^A]\) is the spectrum. Because \(\mathcal {A}\) is a finite dimensional approximation to \(A\), it lies in a smaller space and we must have \(c_1^A\leq \tilde {c}_1^A\) and \(\tilde {c}_2^A\leq c_2^A\). Therefore, the matrix spectrum is contained within the operator spectrum, \([\tilde {c}_1^A,\tilde {c}_2^A] \subset [c_1^A,c_2^A]\).

If \(A^{-1}: X' \rightarrow X\) is the inverse operator of \(A\), its operator spectrum is the reciprocal of the spectrum for \(A\), i.e. \([\frac {1}{c_2^A},\frac {1}{c_1^A}]\). We should have \begin{equation} \frac {1}{c_2^A}\lVert v \rVert _{X'}^{2} \leq \langle A^{-1}v,v \rangle \leq \frac {1}{c_1^A} \lVert v \rVert _{X'}^2 \quad \forall v\in X'. \end{equation} The discrete version is \begin{equation} \tilde {c}_1^{A^{-1}} \lVert \underline {v} \rVert ^2 \leq \underline {v}^{\mathrm {T}} \underline {A}^{-1} \underline {v} \leq \tilde {c}_2^{A^{-1}} \lVert \underline {v} \rVert ^2 \quad \forall \underline {v}\in \mathbb {R}^M. \end{equation} N.B. The discretized matrix \(\underline {A}^{-1}\) for \(A^{-1}\) and the inverse matrix of the discretized matrix \(\mathcal {A}^{-1}\) for \(A\) are two different things. (see Discretization of the inverse of an operator)

Let \(B\) be the preconditioner for \(A\). Its inverse operator \(B^{-1}: X \rightarrow X'\) should be spectrally equivalent to \(A\): \begin{equation} \gamma _1 \langle B^{-1}u,u \rangle \leq \langle Au,u \rangle \leq \gamma _2 \langle B^{-1}u,u \rangle \quad \forall u\in V. \end{equation} If the spectrum of \(B\) is \([c_1^B,c_2^B]\), the spectrum of its inverse is \([\frac {1}{c_2^B},\frac {1}{c_1^B}]\). Then we have \begin{equation} \gamma _1=c_1^A c_1^B, \gamma _2=c_2^A c_2^B. \end{equation}

In the finite dimensional space \(V_h\), the spectral equivalence condition is \begin{equation} \gamma _1 \langle B^{-1}u_{h},u_{h} \rangle \leq \langle Au_{h},u_{h} \rangle \leq \gamma _2 \langle B^{-1}u_{h},u_{h} \rangle \quad \forall u_{h}\in V_{h}. \end{equation} After discretization, the spectral equivalence condition becomes \begin{equation} \gamma _1 \langle \underline {B}^{-1} \underline {u},\underline {u} \rangle \leq \langle \mathcal {A}\underline {u},\underline {u} \rangle \leq \gamma _2 \langle \underline {B}^{-1}\underline {u},\underline {u} \rangle \quad \forall \underline {u}\in \mathbb {R}^N. \end{equation}

According to (Steinbach and Wendland), the spectral equivalence condition determines the condition number of the preconditioned system, which does not depend on the discretization of the system, i.e. both geometric discretization and function discretization. \begin{equation} \kappa (\underline {B}^{-1}\mathcal {A}) \leq \frac {\gamma _2}{\gamma _1} = \frac {c_2^Ac_2^B}{c_1^Ac_1^B}. \end{equation} This is a very important and nice feature, which means when an iterative solver is adopted to solve the preconditioned system, the number of iterations is stable and will not increase with the problem size.

References

   Susanne C. Brenner. The Mathematical Theory of Finite Element Methods. Springer New York, 3 edition. ISBN 978-1-4419-2611-1. URL https://book.douban.com/subject/4219448/#.

   Olaf Steinbach and Wolfgang L. Wendland. The construction of some efficient preconditioners in the boundary element method. 9(1-2):191–216. URL http://link.springer.com/article/10.1023/A:1018937506719.

1Duality pairing \(\langle \cdot ,\cdot \rangle _{X',X}\) is an operation which applies a function/element in the dual space \(X'\) to a function/element in the primal space \(X\). \(X\) is a Banach space and needs not be a Hilbert space. The order of the dual function and primal function in the angle brackets does not matter. When the context is clear, the script \(X',X\) can be omitted.