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.

No abstract uploaded!

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed l_1 (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates l_1 and l_1 norms through a nonnegative parameter l_1 , similar to l_1 with l_1 , and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem and discuss the exact recovery of l_1 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l_1 norm minimal solution if the sensing matrix l_1 satisfies a restricted isometry property (RIP). Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The inner loop concerns an l_1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value l_1 , and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l_1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix l_1 . DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two

The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the immersed boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property.

It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality" in some degree. In this note, we give the error estimate of hyperbolic cross (HC) approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. The exponential convergences in both regular and optimized HC approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Both regular and optimized HC have been investigated with Laguerre-chaos, Charlier-chaos and Hermite-chaos in the numerical experiment. The discussion of the connection between the standard ANOVA approximation and Galerkin approximation is in the appendix.