Quadrature and Schatz’s Pointwise Estimates for Finite Element Methods

We investigate numerical integration effects on weighted pointwise estimates. We prove that local weighted pointwise estimates will hold, modulo a higher order term and a negative-order norm, as long as we use an appropriate quadrature rule. To complete the analysis in an application, we also prove optimal negative-order norm estimates for a corner problem taking into account quadrature. Finally, we present an example to show that our result is sharp. AMS subject classification (2000) 65N15     

Mortar Upwind Finite Volume Element Method with Crouzeix-Raviart Element for Parabolic Convection Diffusion Problems

In this paper,we study the semi-discrete mortar upwind finite volume element method with the Crouzeix-Raviart element for the parabolic convection diffusion problems. It is proved that the semi-discrete mortar upwind finite volume element approximations derived are convergent in the H~1-and L~2-norms.     

Finite Volume Element Predictor-corrector Method for a Class of Nonlinear Parabolic Systems

A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.     

Analysis of Linear Triangular Elements for Convection-diffusion Problems by Streamline Diffusion Finite Element Methods

This paper is devoted to studying the superconvergence of streamline diffusion finite element methods for convection-diffusion problems.In [8],under the condition thatε≤h~2 the optimal finite element error estimate was obtained in L~2-norm.In the present paper,however,the same error estimate result is gained under the weaker condition thatε≤h.     

Domain Decomposition Method Modified by Characteristic Finite Element Procedure for System of Parabolic Equations with Discontinuous Coefficients

In this work, system of parabolic equations with discontinuous coefficients is studied. The domain decomposition method modified by a characteristic finite element procedure is applied. A function is defined to approximate the fluxes on inner boundaries by using the solution at the previous level. Thus the parallelism is achieved. Convergence analysis and error estimate are also presented.     

Jensen’s Inequality for Backward Stochastic Differential Equations

Abstract Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g- expectation in [4, 7–9]. *Project supported by the National Natural Science Foundation of China (No.10325101) and the Science Foundation of China University of Mining and Technology.     

Quantum exotic PDE’s

Following the previous works on the Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley–Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally, the extension of the concept of exotic PDE’s, recently introduced by Prástaro, has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by Prástaro.     

Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system

We rigorously derive energy estimates for the second order vector wave equation with gauge condition for the electric field with non-constant electric permittivity function. This equation is used in the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system. Our numerical experiments illustrate efficiency of the modified hybrid scheme in two and three space dimensions when the method is applied for generation of backscattering data in the reconstruction of the electric permittivity function.     

A Nonconforming Arbitrary Quadrilateral Finite Element Method for Approximating Maxwell＇s Equations

The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes. The error estimates are derived, which are the same as those for conforming elements under conventional regular meshes.     

On Jamet’s Estimates for the Finite Element Method with Interpolation at Uniform Nodes of a Simplex

We suggest a new geometric characteristic of a simplex. This characteristic tends to zero together with the characteristic introduced by Jamet in 1976. Jamet’s characteristic was used in upper estimates for the error of approximation of the derivatives of a function on a simplex by the corresponding derivatives of the polynomial interpolating the values of the function at uniform nodes of the simplex. The use of our characteristic for controlling the form of an element of a triangulation allows us to perform a small finite number of operations. We present an example of a function with lower estimates for approximation of the uniform norms of the derivatives by the corresponding derivatives of the Lagrange interpolating polynomial of degree n. This example shows that, for a broad class of d-simplices, Jamet’s estimates cannot be improved on the set of functions under consideration. On the other hand, for d = 3 and n = 1, we present an example showing that, in general, Jamet’s estimates can be improved.     

Galerkin Methods for Coupled Hyperbolic and Parabolic Linear System

In this paper, we consider the Galerkin methods for the coupled hyperbolic and parabolic linear system and obtain a priori error estimates for semi-discrete and fully discrete Galerkin approximations on the basis of paper [1], but we release the restriction of weak coupled action of the solution [1]. §1 Introduction In [1], we considered the coupled system arising from thermo-elastic problems     

Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations

Potential Analysis - We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range $\frac...     

Local and Parallel Finite Element Algorithms for Eigenvalue Problems

Abstract Some new local and parallel finite element algorithms are proposed and analyzed in this paper foreigenvalue problems.With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced tothe solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraicsystems on fine grid by using some local and parallel procedure.A theoretical tool for analyzing these algorithmsis some local error estimate that is also obtained in this paper for finite element approximations of eigenvectorson general shape-regular grids.     

Finite Element Differential–Algebraic Systems for Eddy Current Problems

Finite element discretization of some time-dependent eddy current problems yields ordinary differential–algebraic systems with large sparse matrices. Properties and stability of these systems are analyzed for two classes of eddy current problems: (i) two-dimensional coupled field-circuit problems with arbitrary external circuit connections between conductors; (ii) 2.5-dimensional problems characterized by axisymmetric geometry and non-axisymmetric excitation. Extension of the analysis to many other formulations of eddy current problems in 2D and 3D is straightforward.     

Analysis on a Finite Volume Element Method for Stokes Problems

Finite volume element method for the Stokes problem is considered. We use a conforming piecewise linear function on a fine grid for velocity and piecewise constant element on a coarse grid for pressure. For general triangulation we prove the equivalence of the finite volume element method and a saddle-point problem, the inf-sup condition and the uniqueness of the approximation solution. We also give the optimal order H^1 norm error estimate. For two widely used dual meshes we give the L^2 norm error estimates, which is optimal in one case and quasi-optimal in another ease. Finally we give a numerical example.     

On Harnack’s Inequality for the Linearized Parabolic Monge-Ampère Equation

It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampère quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampère equation under minimal geometric assumptions.     

Viscosity Solutions of Monotonic Functional Parabolic PDE

In this paper,by the technique of coupled solutions,the notion of viscosity solution is ex-tended to quasi-monotonic fully nonlinear parabolic equations with delay,which involves many modelsarising from optimal control theory,economy and finance,biology etc.The comparison,existence anduniqueness are proved.And the results are applied to the retarded Bellman equations.     

A Finite Element Method for Singularly Perturbed Reaction—diffusion Problems

A finite element method is proposed for the sing ularly perturbed reaction-diffusion problem.An optimal error bound is derived,independent of the perturbation parameter.     

A Nonconforming Arbitrary Quadrilateral Finite Element Method for Approximating Maxwell's Equations

The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes.The error estimates are derived,which are the same as those for conforming elements under conventional regular meshes.