New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h ²) or \(O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)\) in the sense of L ²-norm or L ^∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.     

Spatial High Accuracy Analysis of FEM for Two-dimensional Multi-term Time-fractional Diffusion-wave Equations

In this paper, high-order numerical analysis of finite element method (FEM) is presented for two-dimensional multi-term time-fractional diffusion-wave equation (TFDWE). First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H¹?norm and temporal convergence in L²-norm with order \(O(h^{2}+\tau^{3-\alpha})\), where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H¹-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.     

An <Emphasis Type="Italic">h</Emphasis>-<Emphasis Type="Italic">p</Emphasis> version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations

We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L ²-, H ¹- and L ^∞-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.     

Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method

Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h~(1+min){α,1}) is established for both the displacement approximation in H~1-norm and the stress approximation in L~2-norm under a mesh assumption, where α 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.     

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L²-norm to the L^p-norm with \({p\geq1}\) and the time-decay rates of the momentums m_i(i = 1,2) in the L²-norm to the L^p-norm with p > 1 and all of the decay rates here are optimal.     

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L _∞(0, T; L ₂(Ω))-norm and to ?u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h ^k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h ^k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

Convergence of FBP algorithm for tomography

Filter back-projection (FBP) algorithms are available and extensively used methods for tomography. In this paper, we prove the convergence of FBP algorithms at any continuous point of image function, in L ²-norm and L ¹-norm under the certain assumptions of image and window functions of FBP algorithms.     

Stabilized finite element method for the viscoelastic Oldroyd fluid flows

In this paper, a new stabilized finite element method based on two local Gauss integrations is considered for the two-dimensional viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newtonian fluid flows. This new stabilized method presents attractive features such as being parameter-free, or being defined for non-edge-based data structures. It confirms that the lowest equal-order P ₁???P ₁ triangle element and Q ₁???Q ₁ quadrilateral element are compatible. Moreover, the long time stabilities and error estimates for the velocity in H ¹-norm and for the pressure in L ²-norm are obtained. Finally, some numerical experiments are performed, which show that the new method is applied to this model successfully and can save lots of computational cost compared with the standard ones.     

Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by \(Pf(x) = {\sum }_{y} p(x, y)f(y)\). We assume that the kernels p_n(x, y) associated to Pⁿ satisfy Gaussian upper bounds but do not assume they satisfy the Hölder continuity property and the temporal regularity. Denote by L = I ? P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L^p norm inequalities for the spectral multipliers of L. We also obtain the weighted L^p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

The rate of decay of stable periodic solutions for Duffing equation with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conditions

For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L ^p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L ^p -norm and the \({L^{\infty}}\)-norm.     

Basis Properties of the System of Root Functions of the Sturm–Liouville Operator with Degenerate Boundary Conditions: I

The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L¹(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L²(0, π).     

On Subspaces of Cesàro Spaces

We obtain a characterization of subspaces of L _p , with 1 < p < ∞, on which the L _p -norm is equivalent to the norm of the Cesàro space Ces _p . Also, we show that Ces _p has a complemented copy of the Cesàro sequence space ces _p .     

Cardinal interpolation with general multiquadrics: convergence rates

This article pertains to interpolation of Sobolev functions at shrinking lattices \(h\mathbb {Z}^{d}\) from L _p shift-invariant spaces associated with cardinal functions related to general multiquadrics, ? _{α, c} (x) := (|x|² + c ²) ^α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, L _p error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h ^k ) for functions with derivatives of order up to k in L _p , \(1<p<\infty \)). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.     

Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups

This article completes the authors’s series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness √K ? 1 in the uniform norm and with order of closeness K ? 1 in the Sobolev L _p ¹ -norm for all p < C/K?1. We construct two examples, demonstrating the asymptotic sharpness of our results.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

A result analogous to G. Szego’s theorem

The paper proves a new result similar to the well known theorems of G. Szego and A. Kolmogorov on the best approximation by analytic polynomials in weighted L ^p spaces. Such results are essential in prediction theory for stationary processes. It is well known, that for one step prediction, the size of the best approximation is the same for all p. The paper proves that for two step prediction the best approximation sizes are different for p = 2 and p = ∞.