The Entropy-Ultra-Bee scheme was developed for the linear advection equation and extended to the Euler system of gas dynamics in [13]. It was expected that the technology could be applied only to the second characteristic field of the system and the computation in the other two nonlinear fields would be implemented by the Godunov scheme. However, the numerical experiments in [13] showed that the scheme, though having improved the wave resolution in the second field, produced numerical oscillations in the other two nonlinear fields. Sophisticated entropy increaser was designed to suppress the spurious oscillations by increasing the entropy when there are waves in the two nonlinear fields presented. However, the scheme is then neither efficient nor robust with problem-related parameters. The purpose of this paper is to fix this problem. To this end, we first study a $3 × 3$ linear system and apply the technology precisely to its second characteristic field while maintaining the computation in the other two fields implemented by the Godunov scheme. We then follow the discussion for the linear system to apply the Entropy-Ultra-Bee technology to the second characteristic field of the Euler system in a linearlized field-by-field fashion to develop a modified Entropy-Ultra-Bee scheme for the system. Meanwhile, a remark is given to explain the problem of the previous Entropy-Ultra-Bee scheme in [13]. A reference solution is constructed for computing the numerical entropy, which maintains the feature of the density and flats the velocity and pressure to constants. The numerical entropy is then computed as the entropy cell-average of the reference solution. Several limitations are adopted in the construction of the reference solution to further stabilize the scheme. Designed in such a way, the modified Entropy-Ultra-Bee scheme has a unified form with no problem-related parameters. Numerical experiments show that all the spurious oscillations in smooth regions are gone and the results are better than those of the previous Entropy-Ultra-Bee scheme in [13].

We perform the analysis of the $hp$ finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree $p$ of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the $hp$-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also presented.

The possibly most popular regularization method for solving the least squares problem $\mathop{\rm min}\limits_x$$||Ax-b||_2$ with a highly ill-conditioned or rank deficient coefficient matrix $A$ is the Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when $A$ has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 (2011), 87-103].

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomée for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomée and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics is driven by a $d$-dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [20, 28] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.