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
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
This article develops the primal hybrid finite element method with Lagrange multipliers to approximate nonlinear parabolic initial-boundary value problems with gradient type non-linearity.A modified elliptic projectio...This article develops the primal hybrid finite element method with Lagrange multipliers to approximate nonlinear parabolic initial-boundary value problems with gradient type non-linearity.A modified elliptic projection is used to produce optimal order error estimates for the semi-discrete and backward Euler-based complete discrete schemes.In addition,error estimates in L^(∞)-norm are established which are optimal in nature.Superconvergence result of the gradient in L^(∞)-norm is discussed for the error between the primal hybrid solution and elliptic projection.As a bi-product,the proposed analysis provides optimal error analysis for non-conforming CR-elements.Finally,numerical tests are performed to validate the theoretical findings.展开更多
This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is li...This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.展开更多
The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dim...The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dimension of the finite element space and N is the number of time steps.展开更多
In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L^1 and no growth assumption is made on the lower-order term in divergence form.
.In this paper,a quadratic finite volume method(FVM)for parabolic problems is studied.We first discretize the spatial variables using a quadratic FVM to obtain a semi-discrete scheme.We then employ the backward Euler ....In this paper,a quadratic finite volume method(FVM)for parabolic problems is studied.We first discretize the spatial variables using a quadratic FVM to obtain a semi-discrete scheme.We then employ the backward Euler method and the Crank-Nicolson method respectively to further disctetize the time vatiable so as to derive two full-discrete schemes.The existence and uniqueness of the semi-discrete and full-discrete FVM solutions are established and their optimal error estimates are derived.Finally,we give numerical examples to illustrate the theoretical results.展开更多
Focuses on the formulation of a number of least-squares mixed finite element schemes to solve the initial boundary value problem of a nonlinear parabolic partial differential equation. Convergence analysis; Informatio...Focuses on the formulation of a number of least-squares mixed finite element schemes to solve the initial boundary value problem of a nonlinear parabolic partial differential equation. Convergence analysis; Information on the least-squares mixed element schemes for nonlinear parabolic problem.展开更多
In this paper, we develop the cascadic multigrid method for parabolic problems. The optimal convergence accuracy and computation complexity are obtained.
In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is establishe...In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.展开更多
A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discon...A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.展开更多
The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic prob- lems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyz...The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic prob- lems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method （MG） is greatly reduced. Nu- merical experiments compare the efficiency of Direct Conjugate Gradient Method （DCG） and Extrapolation Cascadic Multigrid Method （EXCMG）. Last, we propose a Time- Extrapolation Algorithm （TEA）, which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG＇s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.展开更多
In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal diff...In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.展开更多
In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouvill...In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.展开更多
This paper deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are construc...This paper deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone weighted average iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.展开更多
This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite diffe...This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite difference scheme for the partial differential equation, we construct a monotone domain decomposition algorithm based on a Schwarz alternating method and a box-domain decomposition. This algorithm needs only to solve linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear discrete problem. domain decomposition algorithm is estimated The rate of convergence of the monotone Numerical experiments are presented.展开更多
In this paper,we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions.The proof is obtained by a bootstrap argument and maximal r...In this paper,we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions.The proof is obtained by a bootstrap argument and maximal regularity estimates.If n≥10/3m,we also give another proof which does not use maximal regularity estimates.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
Monotonicity formulas play a central role in the study of free boundary problems.In this note,we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measure cones.
In the past few years,the number of processor cores of top ranked supercomputers has increased drastically.It is challenging to design efficient parallel algorithms that offer such a high degree of parallelism,especia...In the past few years,the number of processor cores of top ranked supercomputers has increased drastically.It is challenging to design efficient parallel algorithms that offer such a high degree of parallelism,especially for certain time-dependent problems because of the sequential nature of“time”.To increase the degree of parallelization,some parallel-in-time algorithms have been developed.In this paper,we give an overview of some recently introduced parallel-in-time methods,and present in detail the class of space-time Schwarz methods,including the standard and the restricted versions,for solving parabolic partial differentialequations.Some numerical experiments carried out on a parallel computer with a large number of processor cores for three-dimensional problems are given to show the parallel scalability of the methods.In the end of the paper,we provide a comparison of the parallel-in-time algorithms with a traditional algorithm that is parallelized only in space.展开更多
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.
基金support provided by the DST-FIST program(Govt.of India)for setting up the computinglab facility called Center for Mathematical&Financial Computing(C-MFC)at the LNM Institute of Information Technology under the scheme"Fund for Improvement of Science and Technology"(FIST-No.SR/FST/MS-I/2018/24).
文摘This article develops the primal hybrid finite element method with Lagrange multipliers to approximate nonlinear parabolic initial-boundary value problems with gradient type non-linearity.A modified elliptic projection is used to produce optimal order error estimates for the semi-discrete and backward Euler-based complete discrete schemes.In addition,error estimates in L^(∞)-norm are established which are optimal in nature.Superconvergence result of the gradient in L^(∞)-norm is discussed for the error between the primal hybrid solution and elliptic projection.As a bi-product,the proposed analysis provides optimal error analysis for non-conforming CR-elements.Finally,numerical tests are performed to validate the theoretical findings.
基金This work is supported by NSFC(Grant Nos.11771035,11771162,11571128,61473126,91430216,91530204,11372354 and U1530401),a grant from the RGC of HK 11300517,China(Project No.CityU 11302915),China Postdoctoral Science Foundation under grant No.2016M602273,a grant DRA2015518 from 333 High-level Personal Training Project of Jiangsu Province,and the USA National Science Foundation grant DMS-1315259the USA Air Force Office of Scientific Research grant FA9550-15-1-0001.Jiwei Zhang also thanks the hospitality of Hong Kong City University during the period of his visiting.
文摘This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.
文摘The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dimension of the finite element space and N is the number of time steps.
文摘In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L^1 and no growth assumption is made on the lower-order term in divergence form.
基金supported in part by the National Natural Science Foundation of China under grant No.11901506by the Shandong Province Natural Science Foundation under grant No.ZR2018QA003 and ZR2021MA010.
文摘.In this paper,a quadratic finite volume method(FVM)for parabolic problems is studied.We first discretize the spatial variables using a quadratic FVM to obtain a semi-discrete scheme.We then employ the backward Euler method and the Crank-Nicolson method respectively to further disctetize the time vatiable so as to derive two full-discrete schemes.The existence and uniqueness of the semi-discrete and full-discrete FVM solutions are established and their optimal error estimates are derived.Finally,we give numerical examples to illustrate the theoretical results.
基金Major State Basic Research Program of People's Republic of China(G1999032803).
文摘Focuses on the formulation of a number of least-squares mixed finite element schemes to solve the initial boundary value problem of a nonlinear parabolic partial differential equation. Convergence analysis; Information on the least-squares mixed element schemes for nonlinear parabolic problem.
基金Subeidized by the Special Funds for Major State Basic Research Projects.
文摘In this paper, we develop the cascadic multigrid method for parabolic problems. The optimal convergence accuracy and computation complexity are obtained.
基金the National Science Foundation(Grant Nos.DMS0409297,DMR0205232,CCF-0430349)US National Institute of Health-National Cancer Institute(Grant No.1R01CA125707-01A1)+2 种基金the National Natural Science Foundation of China(Grant No.10571172)the National Basic Research Program(Grant No.2005CB321704)the Youth's Innovative Program of Chinese Academy of Sciences(Grant Nos.K7290312G9,K7502712F9)
文摘In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.
基金The authors thank the referees and the editor for their invaluable comments and suggestions which have helped to improved the paper greatly. Also, the authors would like to thank Prof. Xiu Ye for useful discussions and Dr. Paul Scott for helpful revision. This work is done when the first author is visiting Department of Mathematics and Statistics, University of Arkansas at Little Rock under supported by the State Scholarship Fund from the China Scholarship Council. The first author＇s research is partially supported by the Natural Science Foundation of Shandong Province of China grant ZR2013AM023, ZR2012AM019.
文摘A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.
基金This work was supported by the National Natural Science Foundation of China （No. 11071067, 41204082, 11301176）, the Research Fund for the Doctoral Program of Higher Education of China （No. 20120162120036）, the Hunan Provincial Natural Science Foundation of China （No. 14JJ3070） and the Construct Program of the Key Discipline in Hunan Province.
文摘The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic prob- lems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method （MG） is greatly reduced. Nu- merical experiments compare the efficiency of Direct Conjugate Gradient Method （DCG） and Extrapolation Cascadic Multigrid Method （EXCMG）. Last, we propose a Time- Extrapolation Algorithm （TEA）, which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG＇s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.
基金Acknowledgments. The work was supported by the Natural Science Foundation of China （No.11126117）, CAPES and CNPq of Brazil, and the Doctor Fund of Henan Polytechnic Univer- sity （B2012-098）. The author is very grateful to Professor JinYun Yuan for his kind invitation to visit the Universidade Federal do Paran, Brazil.
文摘In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.
文摘In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.
文摘This paper deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone weighted average iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.
文摘This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite difference scheme for the partial differential equation, we construct a monotone domain decomposition algorithm based on a Schwarz alternating method and a box-domain decomposition. This algorithm needs only to solve linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear discrete problem. domain decomposition algorithm is estimated The rate of convergence of the monotone Numerical experiments are presented.
文摘In this paper,we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions.The proof is obtained by a bootstrap argument and maximal regularity estimates.If n≥10/3m,we also give another proof which does not use maximal regularity estimates.
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.
基金The second author was partially supported by National Key R&D Program of China(2021YFA1003001)NSFC 12025109,and the third author was partially supported by NSFC(11521101).
文摘Monotonicity formulas play a central role in the study of free boundary problems.In this note,we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measure cones.
基金supported by the National Key R&D Program of China 2016YFB0200601the Shenzhen basic research grant JCYJ20160331193229720,JCYJ20170307165328836,JCYJ20170818153840322NSFC 11701133,61531166003,11726636.
文摘In the past few years,the number of processor cores of top ranked supercomputers has increased drastically.It is challenging to design efficient parallel algorithms that offer such a high degree of parallelism,especially for certain time-dependent problems because of the sequential nature of“time”.To increase the degree of parallelization,some parallel-in-time algorithms have been developed.In this paper,we give an overview of some recently introduced parallel-in-time methods,and present in detail the class of space-time Schwarz methods,including the standard and the restricted versions,for solving parabolic partial differentialequations.Some numerical experiments carried out on a parallel computer with a large number of processor cores for three-dimensional problems are given to show the parallel scalability of the methods.In the end of the paper,we provide a comparison of the parallel-in-time algorithms with a traditional algorithm that is parallelized only in space.
