BDDC (Balancing Domain Decomposition by Constraints) and FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting) algorithms with adaptively enriched coarse spaces are developed and analyzed for second order elliptic problems with high contrast and random coefficients. Among many approaches to form adaptive coarse spaces, we consider an approach using eigenvectors of generalized eigenvalues problems defined on each subdomain interface, seeMandel and Sousedk(2007), Galvis and Efendiev(2010), Spillane etal.(2011), Spillane etal.(2013), Klawonn etal.(2015).

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a discrete-type flow depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.

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In this paper, we propose and analyze a nodal-continuous and $H^1$-conforming finite element method for the numerical computation of Maxwell's equations, with singular solution in a fractional order Sobolev space $H^r(\Omega)$, where $r$ may take any value in the most interesting interval $(0, 1)$. The key feature of the method is that mass-lumping linear finite element $L^2$ projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of $L^2(\Omega)$ space. We shall use the nodal-continuous linear finite elements, enriched with one element bubble in each element, to approximate the singular and non-$H^1$ solution. Discontinuous and nonhomogeneous media are allowed in the method. Some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method.

Variational models with \ell_1-norm based regularization, in particular total variation (TV) and its variants, have long been known to offer superior image restoration quality, but processing speed remained a bottleneck, preventing their widespread use in the practice of color image processing. In this paper, by extending the grayscale image deblurring algorithm proposed in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, <i>SIAM J. Imaging Sci.</i>, 1 (2008), pp. 248272], we construct a simple and efficient algorithm for multichannel image deblurring and denoising, applicable to both within-channel and cross-channel blurs in the presence of additive Gaussian noise. The algorithm restores an image by minimizing an energy function consisting of an \ell_1-norm fidelity term and a regularization term that can be either TV, weighted TV, or regularization functions based on higher-order derivatives. Specifically, we use a multichannel