EXPLICIT ERROR ESTIMATE FOR THE NONCONFORMING WILSON'S ELEMENT

In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson＇s element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.

Optimality of Solution with Numerical Investigation for Coronavirus Epidemic Model

The novel coronavirus disease,coined as COVID-19,is a murderous and infectious disease initiated from Wuhan,China.This killer disease has taken a large number of lives around the world and its dynamics could not be controlled so far.In this article,the spatio-temporal compartmental epidemic model of the novel disease with advection and diffusion process is projected and analyzed.To counteract these types of diseases or restrict their spread,mankind depends upon mathematical modeling and medicine to reduce,alleviate,and anticipate the behavior of disease dynamics.The existence and uniqueness of the solution for the proposed system are investigated.Also,the solution to the considered system is made possible in a well-known functions space.For this purpose,a Banach space of function is chosen and the solutions are optimized in the closed and convex subset of the space.The essential explicit estimates for the solutions are investigated for the associated auxiliary data.The numerical solution and its analysis are the crux of this study.Moreover,the consistency,stability,and positivity are the indispensable and core properties of the compartmental models that a numerical design must possess.To this end,a nonstandard finite difference numerical scheme is developed to find the numerical solutions which preserve the structural properties of the continuous system.The M-matrix theory is applied to prove the positivity of the design.The results for the consistency and stability of the design are also presented in this study.The plausibility of the projected scheme is indicated by an appropriate example.Computer simulations are also exhibited to conclude the results.

EXPLICIT ERROR ESTIMATES FOR MIXED AND NONCONFORMING FINITE ELEMENTS

In this paper, we study the explicit expressions of the constants in the error estimates of the lowest order mixed and nonconforming finite element methods. We start with an explicit relation between the error constant of the lowest order Raviart-Thomas interpolation error and the geometric characters of the triangle. This gives an explicit error constant of the lowest order mixed finite element method. Furthermore, similar results can be ex- tended to the nonconforming P1 scheme based on its close connection with the lowest order Raviart-Thomas method. Meanwhile, such explicit a priori error estimates can be used as computable error bounds, which are also consistent with the maximal angle condition for the optimal error estimates of mixed and nonconforming finite element methods.

Explicit H1-Estimate for the Solution of the Lamé System with Mixed Boundary Conditions

In this paper we consider the Lamé system on a polygonal convex domain with mixed boundary conditions of Dirichlet-Neumann type.An explicit L2 norm estimate for the gradient of the solution of this problem is established.This leads to an explicit bound of the H1 norm of this solution.Note that the obtained upper-bound is not optimal.

Mixed principal eigenvalues in dimension one

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L2-Poincare inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.

Mixed eigenvalues of discrete p-Laplacian

This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation： the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity （which was known years ago）, as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at presented at the end of Section 2 to infinity is also studied. Two examples are illustrate the value of the investigation.

Rigidity Constraints for Large Mesh Deformation

It is a challenging problem of surface-based deformation to avoid apparent volumetric distortions around largely deformed areas. In this paper, we propose a new rigidity constraint for gradient domain mesh deformation to address this problem. Intuitively the proposed constraint can be regarded as several small cubes defined by the mesh vertices through mean value coordinates. The user interactively specifies the cubes in the regions which are prone to volumetric distortions, and the rigidity constraints could make the mesh behave like a solid object during deformation. The experimental results demonstrate that our constraint is intuitive, easy to use and very effective.

AN EXPONENTIAL WAVE INTEGRATOR PSEUDOSPECTRAL METHOD FOR THE SYMMETRIC REGULARIZED-LONG-WAVE EQUATION

An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.