The distribution for eigenvalues of Schur complement of matrices plays an important role in many mathematical problems. In this paper, we firstly present some criteria for H-matrix. Then as application, for two class matrices whose submatrices are γ-diagonally dominant and product γ-diagonally dominant, we show that the eigenvalues of the Schur complement are located in the Geršgorin discs and the Ostrowski discs of the original matrices under certain conditions.

We propose a new reduced basis element-cum-component mode synthesis approach forparametrized elliptic coercive partial differential equations. In the Offline stage weconstruct a Library of interoperable parametrized reference componentsrelevant to some family of problems; in the Online stage we instantiate andconnect reference components (at ports) to rapidly form and query parametricsystems. The method is based on static condensation at the interdomainlevel, a conforming eigenfunction “port” representation at the interface level, andfinally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at theintradomain level. We show under suitable hypotheses that the RB Schur complement is closeto the FE Schur complement: we can thus demonstrate the stability of the discreteequations; furthermore, we can develop inexpensive and rigorous (system-level) aposteriori error bounds. We present numerical results for model many-parameterheat transfer and elasticity problems with particular emphasis on the Online stage; wediscuss flexibility, accuracy, computational performance, and also the effectivity of thea posteriori error bounds.

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold $\text{Sym(}p\text{,}\,\mathbb{R}{{\text{)}}^{++}}\,\times \,\text{Sym(}q\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ block diagonally embedded in $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\,\le \,2$ or $q\,\le \,2$.

A sparse algebraic multigrid method is studied as a cheap and accurateway to compute approximations of Schur complements of matricesarising from the discretization of some symmetric and positive definitepartial differential operators. The construction of such a multigrid isdiscussed and numerical experiments are used to verify the propertiesof the method.

A Legendre spectral collocation method is presented for the solutionof the biharmonic Dirichlet problem on a square. The solution andits Laplacian are approximated using the set of basis functions suggestedby Shen, which are linear combinations of Legendre polynomials. A Schurcomplement approach is used to reduce the resulting linear system to oneinvolving the approximation of the Laplacian of the solution on the twovertical sides of the square. The Schur complement system is solved bya preconditioned conjugate gradient method. The total cost of the algorithmis O(N 3). Numerical results demonstrate the spectral convergence of the method.