This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the p...This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.展开更多
We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve...We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the one- dimensional one-phase inverse Stefan problem.展开更多
This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incr...This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incremental unknowns methods. Through the stability analyzing for the scheme, it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.展开更多
This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classificati...This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.展开更多
A single step scheme with high accuracy for solving parabolic problem is proposed. It is shown that this scheme possesses good stability and fourth order accuracy with respect to both time and space variables, which a...A single step scheme with high accuracy for solving parabolic problem is proposed. It is shown that this scheme possesses good stability and fourth order accuracy with respect to both time and space variables, which are superconvergent.展开更多
This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admi...This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained.展开更多
We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operat...We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.展开更多
A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element m...A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.展开更多
At recent, Hourgat et gave a domain decomposition algorithm for elliptic problems which can be implemented in parallel. Many numerical experiments have illustrated its efficiency. In the present paper, we apply this a...At recent, Hourgat et gave a domain decomposition algorithm for elliptic problems which can be implemented in parallel. Many numerical experiments have illustrated its efficiency. In the present paper, we apply this algorithm to solve the discrete parabolic problems, analyse its convergence and show that its convergence rale is about (1 - 2p + σp2 ) which is nearly optimal and independent of the parameter τ, where σ τ O((1 +H )(1 + ln(H / h))2 ). 0 【 p 【 1 / σ,τ,h,H are the time step size, finite element parameter and subdomain diameter, respectively.展开更多
Consider the piecewise linear finite element subspace S and parabolic semi discrete Green’s function of gradient type G h(t)∈Sk.The asymptotic optimal estimatedxdt【C|Inh| and two applications are discussed.
In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the...In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nieolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.展开更多
A class of nonlinear neutral partial differential equations,uas considered, and some oscillation criteria for such equations subject to two different boundary value conditions are established.
This paper deals with a parabolic system in a multi dimentional bounded domain ΩR n with the smooth boundary Ω. We discuss an inverse parabolic problem of determining the indirectly measurable internal heat distri...This paper deals with a parabolic system in a multi dimentional bounded domain ΩR n with the smooth boundary Ω. We discuss an inverse parabolic problem of determining the indirectly measurable internal heat distribution at any intermediate moment from the heat distribution measurements in arbitrary accessible subdomain ωΩ at some time interval. Our main result is the Hlder stability estimate in the inverse problem and the proof is completed with a Carleman estimate and a eigenfunction expansion for parabolic equations.展开更多
We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also b...We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation . Using the inverse moments problem techniques we obtain an approximate solution of . Then we find a numerical approximation of when solving the integral equation , because solving the previous integral equation is equivalent to solving the equation .展开更多
The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the sp...The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.展开更多
in this work,we study the quasilinear initial-boundary value problem , where is a system of real smooth vector fields which is defined on an open domain M of R'', and satisfies the Hormanderls condition,.Assu...in this work,we study the quasilinear initial-boundary value problem , where is a system of real smooth vector fields which is defined on an open domain M of R'', and satisfies the Hormanderls condition,.Assume that is non characteristic for the system X,,..',Xm. Under some hypothesis for the boundary of domain and the elliptic structure condition for nonlinear coerfficients Aij, Bj, C,(i, j= 1, ..', m), we have proved that the existence and regularity of solution for aboveinitialboudary value problems.展开更多
In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator wh...In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.展开更多
This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. ...This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.展开更多
The problem is to considerer a parabolic equation depending on a coefficient a (t), and find the solution of the equation and the coefficient. The objective is to solve the problem as an application of the inverse mom...The problem is to considerer a parabolic equation depending on a coefficient a (t), and find the solution of the equation and the coefficient. The objective is to solve the problem as an application of the inverse moment problem. An approximate solution and limits will be found for the error of the estimated solution using the techniques of inverse problem moments. In addition, the method is illustrated with several examples.展开更多
In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.
文摘This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.
文摘We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the one- dimensional one-phase inverse Stefan problem.
文摘This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incremental unknowns methods. Through the stability analyzing for the scheme, it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.SJ08A05
文摘This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.
基金Supported by The National Natural Science Foundations of China (19871027)
文摘A single step scheme with high accuracy for solving parabolic problem is proposed. It is shown that this scheme possesses good stability and fourth order accuracy with respect to both time and space variables, which are superconvergent.
基金Project supported by the NSFC (10971019)Scientific Research Fund of Guangxi Education Department (201012MS067)USM Grant No.12.09.05
文摘This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained.
文摘We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.
文摘A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.
基金The work of the first author is supported by the National Natural Science Foundation of ChinaThe work of the second author is supported by the Natural Science Foundation of Tsinghua University.
文摘At recent, Hourgat et gave a domain decomposition algorithm for elliptic problems which can be implemented in parallel. Many numerical experiments have illustrated its efficiency. In the present paper, we apply this algorithm to solve the discrete parabolic problems, analyse its convergence and show that its convergence rale is about (1 - 2p + σp2 ) which is nearly optimal and independent of the parameter τ, where σ τ O((1 +H )(1 + ln(H / h))2 ). 0 【 p 【 1 / σ,τ,h,H are the time step size, finite element parameter and subdomain diameter, respectively.
基金The project is supported by the National Natural Science Foundation of China.
文摘Consider the piecewise linear finite element subspace S and parabolic semi discrete Green’s function of gradient type G h(t)∈Sk.The asymptotic optimal estimatedxdt【C|Inh| and two applications are discussed.
基金in part supported by the Distinguished Young Scholars Fund of Xinjiang Province(2013711010)NCET-13-0988the NSF of China(11271313,11271298,61163027,and 11362021)
文摘In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nieolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.
文摘A class of nonlinear neutral partial differential equations,uas considered, and some oscillation criteria for such equations subject to two different boundary value conditions are established.
文摘This paper deals with a parabolic system in a multi dimentional bounded domain ΩR n with the smooth boundary Ω. We discuss an inverse parabolic problem of determining the indirectly measurable internal heat distribution at any intermediate moment from the heat distribution measurements in arbitrary accessible subdomain ωΩ at some time interval. Our main result is the Hlder stability estimate in the inverse problem and the proof is completed with a Carleman estimate and a eigenfunction expansion for parabolic equations.
文摘We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation . Using the inverse moments problem techniques we obtain an approximate solution of . Then we find a numerical approximation of when solving the integral equation , because solving the previous integral equation is equivalent to solving the equation .
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.
文摘in this work,we study the quasilinear initial-boundary value problem , where is a system of real smooth vector fields which is defined on an open domain M of R'', and satisfies the Hormanderls condition,.Assume that is non characteristic for the system X,,..',Xm. Under some hypothesis for the boundary of domain and the elliptic structure condition for nonlinear coerfficients Aij, Bj, C,(i, j= 1, ..', m), we have proved that the existence and regularity of solution for aboveinitialboudary value problems.
基金Project supported by National Natural Science Foundation ofChina (Grant No .10471089)
文摘In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.
文摘This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.
文摘The problem is to considerer a parabolic equation depending on a coefficient a (t), and find the solution of the equation and the coefficient. The objective is to solve the problem as an application of the inverse moment problem. An approximate solution and limits will be found for the error of the estimated solution using the techniques of inverse problem moments. In addition, the method is illustrated with several examples.
文摘In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.
