On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given by (1) In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞ ‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian (2) We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞ . We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.     

Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with inhomogeneous boundary conditions

In this article, Richardson extrapolation technique is employed to investigate the local ultraconvergence properties of Lagrange finite element method using piecewise polynomials of degrees () for the second order elliptic problem with inhomogeneous boundary. A sequence of special graded partition are proposed and a new interpolation operator is introduced to achieve order local ultraconvergence for the displacement and derivative.     

Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

A numerical scheme is constructed for the problems in which the diffusion and convection parameters (?₁ and ?₂ , respectively) both are small, and the convection and source terms have a jump discontinuity in the domain of consideration. Depending on the magnitude of the ratios , and two different cases have been considered separately. Through rigorous analysis, the theoretical error bounds on the singular and regular components of the solution are obtained separately, which shows that in both cases the method is convergent uniformly irrespective of the size of the parameters ?₁, ?₂ . Two test problems are included to validate the theoretical results.     

Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary,to a reaction–diffusion equation on the curve

We consider a semidiscrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain , such that the curve meets the boundary orthogonally, and the forcing is a function of the solution of a reaction–diffusion equation that holds on the evolving curve. We prove optimal order error bounds for the resulting approximation and present numerical experiments.     

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

In this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and (ii) the potentials given on , where 0 < α < 1 , respectively. Inverse spectral problems, Sturm–Liouville operator, spectrum, uniqueness.     

Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions

In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has memory requirement and computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations.     

Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

In this paper, we are interested in the long time behavior of approximate solutions to a free boundary model which appears in the modeling of concrete carbonation [1]. In particular, we study the long time regime of the moving interface. The numerical solutions are obtained by an implicit in time and finite volume in space scheme. We show the existence of solutions to the scheme and, following [2‐3], we prove that the approximate free boundary increases in time following a ‐law. Finally, we supplement the study through numerical experiments.     

A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics

In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time-fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative ( D _x and D _t), fractional-order derivative ( and ), fractional-order integration ( and ) and delay terms ( and ) of approximated function u(x,t). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time-fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine-Gordon, parabolic integro-differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi-dimensional physical problems of highly nonlinear fractional or variable order of complex nature.     

Staggered Taylor–Hood and Fortin elements for Stokes equations of pressure boundary conditions in Lipschitz domain

The purpose of this paper is to develop a general theory on how the inf-sup stable and convergent elements of the velocity Dirichlet boundary (VDB)-Stokes problem with no-slip VDB are still inf-sup stable and convergent for the pressure Dirichlet boundary (PDB)-Stokes problem with PDB in Lipschitz domain. The PDB-Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to (H¹(Ω))², sharply in contrast to the VDB-Stokes problem whose velocity solution still belongs to (H¹(Ω))², and unexpectedly, some well-known inf-sup stable and convergent VDB-Stokes elements may or may no longer correctly converge. It turns out that the inf-sup condition of the PDB-Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary. In this paper we propose two families of staggered elements: staggered Taylor–Hood elements with ℓ ≥ 1 (continuous in both velocity and pressure) and staggered Fortin elements with m ≥ 1 (continuous in velocity and discontinuous in pressure) on triangles, for solving the PDB-Stokes problem in Lipschitz domain. We show that the two families are inf-sup stable and are correctly convergent for the non-H¹ singular velocity. Numerical results illustrate the proposed elements and the theoretical results.     

Fast high order difference schemes for the time fractional telegraph equation

In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ 2-1_σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order and , respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ 2-1_σ method.     

Superconvergence analysis of an energy stable scheme with three step backward differential formula-finite element method for nonlinear reaction–diffusion equation

A three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H²-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L²-norm is deduced which help bound the numerical solutions in L^∞-norm. Superconvergent property of in H¹-norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.     

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

In this paper, we consider the numerical solution of the time‐fractional telegraph equation with a nonlocal boundary condition. A novel barycentric Lagrange interpolation collocation method is developed to solve this equation. Two difficulties have been sorted: the singularity of the integration and the higher accuracy. At the same, we put forward a steady barycentric Lagrange interpolation technique to overcome the new “Runge” phenomenon in computation. Error estimates of the barycentric Lagrange interpolation and the time‐fractional telegraph system for the present method are presented in Sobolev spaces. High convergence rates of the proposed method are obtained and are consisted with the numerical values. Especially in the time dimension, we get the error bound, for h‐refinement and for n_t‐density in the L₂ norms. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time‐fractional telegraph equation. Experiments demonstrate the high convergence rates of the proposed method are consisted with the theoretical values.     

H1-superconvergence of finite difference method based on Q1-element on quasi-uniform mesh for the 3D Poisson equation

In this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q₁-element on a quasi-uniform mesh. First, under the regularity assumption of , the H¹-superconvergence of the FD solution u_h based on the Q₁-element to the first-order interpolation function is obtained. Next, the H¹-superconvergence of the second-order interpolation postprocessing function based on the FD solution u_h to u is provided. Finally, numerical tests are presented to show the H¹-superconvergence result of the FD postprocessing function to u if .     

Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations

In this article, a new weighted and compact conservative difference scheme for the symmetric regularized long wave (SRLW) equations is considered. The new scheme is decoupled and linearized in practical computation, that is, at each time step only two tridiagonal systems of linear algebraic equations need to be solved. It is proved by the discrete energy method that the compact scheme is uniquely solvable, the convergence and stability of the difference scheme are obtained, and its numerical convergence order is in the ‐norm. Numerical experiment results show that the scheme is efficient and reliable.     

Numerical method for generalized time fractional KdV-type equation

In this article, an efficient numerical method for linearized and nonlinear generalized time-fractional KdV-type equations is proposed by combining the finite difference scheme and Petrov–Galerkin spectral method. The scale and weight functions involved in generalized fractional derivative cause too much difficulty in discretization and numerical analysis. Fortunately, motivated by finite difference method for fractional differential equation on graded mesh, the stability and convergence of the constructed method are established rigorously. It is proved that the full discretization schemes of generalized time-fractional KdV-type equation is unconditionally stable in linear case. While for nonlinear case, it is stable under a CFL condition and for not small ϵ, coefficient of the high-order spatial differential term. In addition, the full discretization schemes with respect to linear and nonlinear cases respectively converge to the associated exact solutions with orders and , where τ, α, N and m accordingly indicate the time step size, the order of the fractional derivative, polynomial degree, and regularity of the exact solution. Numerical experiments are carried out to support the theoretical results.     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

We consider a time‐dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω_?, with n = 3,4. The fluid flows in a domain containing a periodical set of “obstacles” (Ω\Ω_?) placed along an inner (n ? 1)‐dimensional manifold . The size of the obstacles is much smaller than the size of the characteristic period ?. An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function σ of the concentration and a large adsorption parameter. The “critical adsorption parameter” depends on the size of the obstacles , and, for different sizes, we derive the time‐dependent homogenized models. These models contain a “strange term” in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of σ. The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3.     

Superconvergence analysis of a nonconforming finite element method for monotone semilinear elliptic optimal control problems

A nonconforming finite element method (FEM) is proposed for optimal control problems (OCPs) governed by monotone semilinear elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection of the adjoint state, respectively. Some global supercloseness and superconvergence estimates are achieved by making full use of the distinguish characters of this element, such as the interpolation equals to its Ritz projection, and the consistency error is 1 − ε ( is small enough) order higher than its interpolation error in the broken energy norm when the exact solution belongs to H^{3 − ε}(Ω). Finally, some numerical results are presented to verify the theoretical analysis.     

A C0‐weak Galerkin finite element method for fourth‐order elliptic problems

This article proposes and analyzes a C⁰‐weak Galerkin (WG) finite element method for solving the biharmonic equation in two‐dimensional and three‐dimensional. The new WG method uses continuous piecewise‐polynomial approximations of degree for the unknown u and discontinuous piecewise‐polynomial approximations of degree k for the trace of on the interelement boundaries. Optimal error estimates are obtained in H², H¹, and L² norms. Numerical experiments illustrate and confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1090–1104, 2016     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.