A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+
t2)is obtained,which is verified by the numerical results.

Quadratic Finite Volume Element Schemes over Triangular Meshes for a Nonlinear Time-Fractional Rayleigh-Stokes Problem

In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.

A Coercivity Result of Quadratic Finite Volume Element Schemes over Triangular Meshes

In this work,we study the coercivity of a family of quadratic finite volume element(FVE)schemes over triangular meshes for solving elliptic boundary value problems.The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as H^(1) and L^(2) error estimates.The novelty of this paper is that,each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts:the first part is the element stiffness matrix of the standard quadratic finite element method(FEM),the second part is the difference between the FVE and FEM on the element boundary,while the third part can be expressed as the tensor product of two vectors.As a result,we reach a sufficient condition to guarantee the existence,uniqueness and coercivity result of the FVE solution on general triangular meshes.Moreover,based on this sufficient condition,some minimum angle conditions with simple,analytic and computable expressions are obtained.By comparison,the existing minimum angle conditions were obtained numerically from a computer program.Theoretical findings are conformed with the numerical results.

A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.

OPTIMAL QUADRATIC NITSCHE EXTENDED FINITE ELEMENT METHOD FOR INTERFACE PROBLEM OF DIFFUSION EQUATION

In this paper, we study Nitsche extended finite element method （XFEM） for the inter- face problem of a two dimensional diffusion equation. Specifically, we study the quadratic XFEM scheme on some shape-regular family of grids and prove the optimal convergence rate of the scheme with respect to the mesh size. Main efforts are devoted onto classifying the cases of intersection between the elements and the interface and prove a weighted trace inequality for the extended finite element functions needed, and the general framework of analysing XFEM c^n be implemented then.

CONSTRUCTIONOFACLASSOFPNSEQUENCESWITHPRIMENUMBERPERIODSOF4t＋1FORM

A class of new PN sequence with prime number periods of 4t +1 form (t is an integer)is constructed.The advantage of these PN sequencs over the m(M) sequence is their large number of alternative periods.They have good pseudo random characteristics demonstrated by the expression of periodic autocorrelation function found out in this paper.

A Non-Singular Boundary Element Method for Interactions between Acoustical Field Sources and Structures

Localized point sources(monopoles)in an acoustical domain are implemented to a three dimensional non-singular Helmholtz boundary element method in the frequency domain.It allows for the straightforward use of higher order surface elements on the boundaries of the problem.It will been shown that the effect of the monopole sources ends up on the right hand side of the resulting matrix system.Some carefully selected examples are studied,such as point sources near and within a concentric spherical core-shell scatterer(with theoretical verification),near a curved focusing surface and near a multi-scale and multi-domain acoustic lens.

A CONFORMING QUADRATIC POLYGONAL ELEMENT AND ITS APPLICATION TO STOKES EQUATIONS

In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,two stable discretizations for the Stokes equations are developed,which can be viewed as the extensions of the P2-P0 and the Q2-(discontinuous)P1 elements,respectively,to polygonal meshes.Numerical results are presented,which support our theoretical claims.

Two-Level Hierarchical PCG Methods for the Quadratic FEM Discretizations of 2D Concrete Aggregate Models

The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases:aggregates which randomly distributed in different shapes,cement paste and internal transition zone(ITZ).Because of different shapes of aggregate and thin ITZs,a huge number of elements are often used in the finite element(FEM)analysis.In order to ensure the accuracy of the numerical solutions near the interfaces,we need to use higher-order elements.The widely used FEM softwares such as ANSYS and ABAQUS all provide the option of quadratic elements.However,they have much higher computational complexity than the linear elements.The corresponding coefficient matrix of the system of equations is a highly ill-conditioned matrix due to the large difference between three phase materials,and the convergence rate of the commonly used solving methods will deteriorate.In this paper,two types of simple and efficient preconditioners are proposed for the system of equations of the concrete aggregate models on unstructured triangle meshes by using the resulting hierarchical structure and the properties of the diagonal block matrices.The main computational cost of these preconditioners is how to efficiently solve the system of equations by using linear elements,and thus we can provide some efficient and robust solvers by calling the existing geometric-based algebraic multigrid(GAMG)methods.Since the hierarchical basis functions are used,we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types,and the grid transfer operators are also trivial.This makes it easy to find the linear element matrix derived directly from the fine level matrix,and thus the overall efficiency is greatly improved.The numerical results have verified the efficiency of the resulting preconditioned conjugate gradient(PCG)methods which are applied to the solution of several typical aggregate models.