Two guaranteed equilibrated error estimators for Harmonic formulations in eddy current problems

In this paper a guaranteed equilibrated error estimator is developed for the 3D harmonic magnetodynamic problem of Maxwell’s system. This system is recasted in the classical $\mathbf{A}?\phi$ potential formulation and solved by the Finite Element method. The error estimator is built starting from the $\mathbf{A}?\phi$ numerical solution by a local flux reconstruction technique. Its equivalence with the error in the energy norm is established. A comparison of this estimator with an equilibrated error estimator already developed through a complementary problem points out the advantages and drawbacks of these two estimators. In particular, an analytical benchmark test illustrates the obtained theoretical results and a physical benchmark test shows the efficiency of these two estimators.     

Quasi gradient-based inversion-free iterative algorithm for solving a class of the nonlinear matrix equations

Inspired by the gradient-based and inversion-free iterations, a new quasi gradient-based inversion-free iterative algorithm is proposed for solving the nonlinear matrix equation $X+A^{\mathrm{T}}{X}^{?n}A=I$. The convergence proof of the suggested algorithm is given. Several matrix norm inequalities are established to depict the convergence properties of this algorithm. Three numerical examples are given to illustrate the effectiveness of the suggested algorithms.     

A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations

The construction of finite element approximations in $\mathbf{H}(\text{div},\Omega )$ usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region $\Omega$. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide $L^2$-errors with accuracy of order $k+1$ for sufficiently smooth flux functions, and of order $r+1$ for flux divergence. An example is $R{T}_k$ spaces on quadrilateral meshes, where $r=k$ or $k?1$ if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy $r+n+1$ as desired, for any $n\ge 1$. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level $k+n$, while keeping the original border fluxes at level $k$. The case $n=1$ has been discussed in the mentioned publication for two particular examples. General stronger enrichment $n>1$ shall be analyzed and applied to Darcy’s flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme.     

Global well-posedness of the d-D Oldroyd-B type models with fractional Laplacian dissipation

This paper focuses on the Cauchy problem of the d-dimensional incompressible Oldroyd-B type models for viscoelastic flow with fractional Laplacian dissipation, namely, with ${(?\Delta )}^{{\eta }_1}u$ and ${(?\Delta )}^{{\eta }_2}\tau$. For ${\eta }_1\ge \frac{1}{2}+\frac{d}{4}$, ${\eta }_2>0$ and ${\eta }_1+{\eta }_2\ge 1+\frac{d}{2}$, we obtain the global regularity of strong solutions when the initial data $(u_0,{\tau }_0)$ are sufficiently smooth.     

Ground state solutions of Pohoz̆aev type for fractional Choquard equations with general nonlinearities

In this paper, we study the fractional Choquard equation

${(?\Delta )}^{s}u+u=\left(\right|x{|}^{?\mu }1F\left(u\right))f\left(u\right),\text{in}{\mathbb{R}}^N,$

where $N\ge 3$, $0<s<1$, $0<\mu <min\{N,4s\}$, and $f\in C(\mathbb{R},\mathbb{R})$ satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz?aev type for the above equation. The result improves some ones in Shen et al. (2016).     

Stability and convergence of semi-implicit time-stepping algorithm for stationary incompressible magnetohydrodynamics

For the stationary incompressible magnetohydrodynamics (MHD) equations, we provide a new uniqueness assumption (A0) and show the exponential stability of the solution. Then, the semi-implicit time-stepping algorithm is used to solve the stationary MHD equations. The algorithm is proved to be unconditionally stable. The discrete velocity and magnetic field are bounded in $L^{\infty }(0,+\infty ;{\mathit{L}}^{\infty }\left(\Omega \right))$ for any space and time step sizes. The error estimates for the algorithm are deduced under the uniqueness conditions. Finally, numerical experiments are performed to testify our theoretical analysis.     

On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |u(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

where $0<q<1$, $i^q$ is the principal value of $i^q$, $D_{q|t}$ is the $q$-derivative with respect to $t$, $\Delta$ is the Laplacian operator in ${\mathbb{R}}^N$, $\lambda \in ??\left\{0\right\}$, $p>1$, and $u(t,x)$ is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |v(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

$i^q{D}_{q|t}v(t,x)+\Delta v(qt,x)=\lambda |u(qt,x){|}^m,t>0,x\in {\mathbb{R}}^N,$     

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

This paper is concerned with the following linearly coupled fractional Kirchhoff-type system

$\left\{\begin{array}{c}\left(a+b{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x\right){(?△)}^{\alpha }u+\lambda u=f\left(u\right)+\gamma v,\text{in}{\mathbb{R}}^3,\hfill \\ \left(c+d{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}v{|}^2\mathrm{d}x\right){(?△)}^{\alpha }v+\mu v=g\left(v\right)+\gamma u,\text{in}{\mathbb{R}}^3,\hfill \\ u,v\in H^{\alpha }\left({\mathbb{R}}^3\right),\hfill \end{array}\right.$

where $a,c,\lambda ,\mu >0$, $b,d\ge 0$ are constants, $\alpha \in [\frac{3}{4},1)$ and $\gamma >0$ is a coupling parameter. Under the general Berestycki–Lions conditions on the nonlinear terms $f$ and $g$, we prove the existence of positive vector ground state solutions of Poho?aev type for the above system via variational methods. Moreover, the asymptotic behavior of these solutions as $\gamma \to 0^{+}$ is explored as well. Recent results from the literature are generally improved and extended.     

Existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff system involving the fractional p-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional $p$-Laplacian

$\begin{array}{c}\hfill \left\{\begin{array}{c}M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)?u\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p?2}u=F_u(x,u,v)+\lambda g\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|v\left(x\right)?v\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|v\right|}^{p?2}v=F_v(x,u,v)+\lambda h\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ u\left(x\right)\to 0,v\left(x\right)\to 0,\text{as}\left|x\right|\to +\infty ,\hfill \end{array}\right.\end{array}$

where ${(?\Delta )}_p^s$ denotes the fractional $p$-Laplacian of order $s\in (0,1)$, $2\le p<\infty$, $ps<N$, $F_u=\frac{?F}{?u}$, $F_v=\frac{?F}{?v}$, $V\left(x\right)$ is allowed to be sign-changing, $\lambda >0$ and $g,h:{\mathbb{R}}^N\to \mathbb{R}$ is a perturbation. Under some certain assumptions on $f$, we obtain the existence of multiple solutions for this problem via Ekeland’s variational principle and mountain pass theorem.     

An accelerated cyclic-reduction-based solvent method for solving quadratic eigenvalue problem of gyroscopic systems

The quadratic eigenvalue problem (QEP) $({\lambda }^{2}M+\lambda G+K)x=0$, with $M^T=M$ being positive definite, $K^T=K$ being negative definite and $G^T=?G$, is associated with gyroscopic systems. In Guo (2004), a cyclic-reduction-based solvent (CRS) method was proposed to compute all eigenvalues of the above mentioned QEP. Firstly, the problem is converted to find a suitable solvent of the quadratic matrix equation (QME) $M{X}^2+GX+K=0$. Then using a Cayley transformation and a proper substitution, the QME is transformed into the nonlinear matrix equation (NME) $Z+A^T{Z}^{?1}A=Q$ with $A=M+K+G$ and $Q=2(M?K)$. The problem finally can be solved by applying the CR method to obtain the maximal symmetric positive definite solution of the NME as long as the QEP has no eigenvalues on the imaginary axis or for some cases where the QEP has eigenvalues on the imaginary axis. However, when all eigenvalues of the QEP are far away from or near the origin, the Cayley transformation seems not to be the best one and the convergence rate of the CRS method proposed in Guo (2004) might be further improved. In this paper, inspired by using a doubling algorithm to solve the QME, we use a Möbius transformation instead of the Cayley transformation to present an accelerated CRS (ACRS) method for solving the QEP of gyroscopic systems. In addition, we discuss the selection strategies of optimal parameter for the ACRS method. Numerical results demonstrate the efficiency of our method.     

GPU implementation of Borůvka’s algorithm to Euclidean minimum spanning tree based on Elias method

We present both sequential and data parallel approaches to build hierarchical minimum spanning forest (MSF) or tree (MST) in Euclidean space (EMSF/EMST) for applications whose input $N$ points are uniformly or boundedly distributed in Euclidean space. Each iteration of the sequential approach takes $O\left(N\right)$ time complexity through combining Borůvka’s algorithm with an improved component-based neighborhood search algorithm, namely sliced spiral search, which is a newly proposed improvement to Bentley’s spiral search for finding a component graph’s closest outgoing point on the plane. It works based on the uniqueness property in Euclidean space, and allows $O\left(1\right)$ time complexity for one search from a query point to find the component’s closest outgoing point at different iterations of Borůvka’s algorithm. The data parallel approach includes a newly proposed two-direction breadth-first search (BFS) implementation on graphics processing unit (GPU) platform, which is specialized for selecting a spanning tree’s shortest outgoing edge. This GPU two-direction parallel BFS enables a tree traversal operation to start from any one of its vertex acting as root. These GPU parallel implementations work by assigning $N$ threads with one thread associated to one input point, one thread occupies $O\left(1\right)$ local memory and the whole algorithm occupies $O\left(N\right)$ global memory. Experiments are conducted on both uniformly distributed data sets and TSPLIB database. We evaluate computation time of the proposed approaches on more than 80 benchmarks with size $N$ growing up to $10^6$ points on personal laptop.     

A fast singular boundary method for 3D Helmholtz equation

This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of $O\left(N^3\right)$ with direct solvers or $O\left(N^2\right)$ with iterative solvers, as well as the memory requirement of $O\left(N^2\right)$. To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from $O\left(N^2\right)$ to $O\left(NlogN\right)$ or $O\left(N\right)$. Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.     

Finite-time blow-up of solutions to a cancer invasion mathematical model with haptotaxis effects

This paper considers the nonlinear system of cancer invasion model with haptotaxis effects in a bounded domain $\Omega ?{\mathbb{R}}^n$ under the homogeneous Neumann and Robin type boundary conditions. This model also includes the growth and decay effects of cancer cells, normal cells and matrix degrading enzymes. This study devoted to estimate the lower bounds for the finite time blow up of non-negative solutions of the given cancer invasion system using first order differential inequality techniques and certain inequalities.