In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is L2 stable. When the finite element space consists of interpolative polynomials of degrees k, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of . Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Goal of this paper is to suitably combine a model with an anisotropic mesh adaptation for the numerical simulation of nonlinear advection-diffusion-reaction systems and incompressible flows in ecological and environmental applications. Using the reduced-basis method terminology, the proposed approach leads to a noticeable computational saving of the online phase with respect to the resolution of the reference model on nonadapted grids. The search of a suitable adapted model/mesh pair is to be meant, instead, in an offline fashion.

In this paper, we investigate the superconvergence property and the L∞-error estimates of mixed finite element methods for a semilinear elliptic control problem. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive some superconvergence results for the control variable. Moreover, we derive L∞-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

In this paper, we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in [Kang & March, IEEE T. Image Process., 16 (2007), 2251-2261]. We use fast algorithms to solve the subproblems, such as the dual projection methods, primal-dual methods and augmented Lagrangian methods. With a special penalization term, some special algorithms are presented. Numerical experiments are given to demonstrate the performance of the proposed methods. We successfully show that our algorithms are effective and efficient due to two reasons: the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately (even 2 inner iterations of the subproblem are enough). It is also observed that better PSNR values are produced using the new algorithms.

The inverse problem considered in this paper is to determine the shape and the impedance of crack from a knowledge of the time-harmonic incident field and the corresponding far field pattern of the scattered waves in two-dimension. The combined single- and double-layer potential is used to approach the scattered waves. As an important feature, this method does not require the solution of u and ∂u/∂v at each iteration. An approximate method is presented and the convergence of this method is proven. Numerical examples are given to show that this method is both accurate and simple to use.

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

This paper proposes a novel multi-scale fluid flow data assimilation approach, which integrates and complements the advantages of a Bayesian sequential assimilation technique, the Weighted Ensemble Kalman filter (WEnKF) [27]. The data assimilation proposed in this work incorporates measurement brought by an efficient multiscale stochastic formulation of the well-known Lucas-Kanade (LK) estimator. This estimator has the great advantage to provide uncertainties associated to the motion measurements at different scales. The proposed assimilation scheme benefits from this multi-scale uncertainty information and enables to enforce a physically plausible dynamical consistency of the estimated motion fields along the image sequence. Experimental evaluations are presented on synthetic and real fluid flow sequences.

Two algorithms for constructing a class of compactly supported conjugate symmetric complex tight wavelet frames ψ = {ψ1, ψ2} are derived. Firstly, a necessary and sufficient condition for constructing the conjugate symmetric complex tight wavelet frames is established. Secondly, based on a given conjugate symmetric low pass filter, a description of a family of complex wavelet frame solutions is provided when the low pass filter is of even length. When one wavelet is conjugate symmetric and the other is conjugate antisymmetric, the two wavelet filters can be obtained by matching the roots of associated polynomials. Finally, two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames which have some vanishing moments.

We present a hybrid numerical method for simulating fluid flow through a compliant, closed tube, driven by an internal source and sink. Fluid is assumed to be highly viscous with its motion described by Stokes flow. Model geometry is assumed to be axisymmetric, and the governing equations are implemented in axisymmetric cylindrical coordinates, which capture 3D flow dynamics with only 2D computations. We solve the model equations using a hybrid approach: we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink, with each part handled separately by means of an appropriate method. Because the singularly-supported surface forcings yield an unsmooth solution, that part of the solution is computed using the immersed interface method. Jump conditions are derived for the axisymmetric cylindrical coordinates. The velocity due to the source and sink is calculated along the tubular surface using boundary integrals. Numerical results are presented that indicate second-order accuracy of the method.

In this paper, we propose the Laguerre spectral method for high order problems with mixed inhomogeneous boundary conditions. It is also available for approximated solutions growing fast at infinity. The spectral accuracy is proved. Numerical results demonstrate its high effectiveness.

In [35,36], we presented an h-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws. A tree data structure (binary tree in one dimension and quadtree in two dimensions) is used to aid storage and neighbor finding. Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled “children”. Extensive numerical tests indicate that the proposed h-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities. In this paper, we apply this h-adaptive method to solve Hamilton-Jacobi equations, with an objective of enhancing the resolution near the discontinuities of the solution derivatives. One- and two-dimensional numerical examples are shown to illustrate the capability of the method.

In this paper, a fast algorithm for Euler’s elastica functional is proposed, in which the Euler’s elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated.

In this paper we extend the source transfer domain decomposition method (STDDM) introduced by the authors to solve the Helmholtz problems in two-layered media, the Helmholtz scattering problems with bounded scatterer, and Helmholtz problems in 3D unbounded domains. The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The details of STDDM is given for each extension. Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper.

In this paper, we propose a robust finite volume scheme to numerically solve the shallow water equations on complex rough topography. The major difficulty of this problem is introduced by the stiff friction force term and the wet/dry interface tracking. An analytical integration method is presented for the friction force term to remove the stiffness. In the vicinity of wet/dry interface, the numerical stability can be attained by introducing an empirical parameter, the water depth tolerance, as extensively adopted in literatures. We propose a problem independent formulation for this parameter, which provides a stable scheme and preserves the overall truncation error of . The method is applied to solve problems with complex rough topography, coupled with h-adaptive mesh techniques to demonstrate its robustness and efficiency.

We present an efficient implementation of volumetric anisotropic image diffusion filters on modern programmable graphics processing units (GPUs), where the mathematics behind volumetric diffusion is effectively reduced to the diffusion in 2D images. We hereby avoid the computational bottleneck of a time consuming eigenvalue decomposition in ℝ3. Instead, we use a projection of the Hessian matrix along the surface normal onto the tangent plane of the local isodensity surface and solve for the remaining two tangent space eigenvectors. We derive closed formulas to achieve this and prevent the GPU code from branching. We show that our most complex volumetric anisotropic diffusion filters gain a speed up of more than 600 compared to a CPU solution.

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive L∞- and H−1 -error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.