In Hashim and Harfash(Appl.Math.Comput.2021),using a finite element method,the attraction-repulsion chemotaxis model(P)in space is discretised;finite differences were used to do the same in time.Furthermore,the existe...In Hashim and Harfash(Appl.Math.Comput.2021),using a finite element method,the attraction-repulsion chemotaxis model(P)in space is discretised;finite differences were used to do the same in time.Furthermore,the existence of a global weak solution to the system(PΔt M)was demonstrated by means of analysis of the convergence of the fully discrete approximate problem(P h,Δt M, ).Moreover,the functions{U,Z,V}were proved to represent a global weak solution to the system(PΔt M)by means of a passage to the limit ,h→0 of the approximate system.This paper’s purpose is to demonstrate that the solutions can be bounded,independent of M.The analysis contained in this paper illustrates the idea of the existence of weak solutions to the model(P),that requires passing to the limits,Δt→0+and M→∞.The time stepΔt is subsequently linked to the cutoff parameter M>1 by positing a demand thatΔt=o(M−1),as M→∞,with the result that the cutoff parameter becomes the only parameter in the problem(PΔt M).The solutions can be bounded,independ-ent of M,with the use of special energy estimates,as demonstrated herein.Then,these M-independent bounds on the relative entropy are employed with the purpose of deriving M-independent bounds on the time-derivatives.Additionally,compactness arguments were utilised to explore the convergence of the finite element approximate problem.The conclu-sion was that a weak solution for(P)existed.Finally,we introduced the error estimate and the implicit scheme was used to perform simulations in one and two space dimensions.展开更多
In this paper,a finite element scheme for the attraction-repulsion chemotaxis model is analyzed.We introduce a regularized problem of the truncated system.Then we obtain some a priori estimates of the regularized func...In this paper,a finite element scheme for the attraction-repulsion chemotaxis model is analyzed.We introduce a regularized problem of the truncated system.Then we obtain some a priori estimates of the regularized functions,independent of the regularization parameter,via deriving a well-defined entropy inequality of the regularized problem.Also,we propose a practical fully discrete finite element approximation of the regularized problem.Next,we use a fixed point theorem to show the existence of the approximate solutions.Moreover,a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived.In addition,the uniqueness of the fully discrete approximations is preformed.Finally,we discuss the convergence to the fully discrete problem.展开更多
文摘In Hashim and Harfash(Appl.Math.Comput.2021),using a finite element method,the attraction-repulsion chemotaxis model(P)in space is discretised;finite differences were used to do the same in time.Furthermore,the existence of a global weak solution to the system(PΔt M)was demonstrated by means of analysis of the convergence of the fully discrete approximate problem(P h,Δt M, ).Moreover,the functions{U,Z,V}were proved to represent a global weak solution to the system(PΔt M)by means of a passage to the limit ,h→0 of the approximate system.This paper’s purpose is to demonstrate that the solutions can be bounded,independent of M.The analysis contained in this paper illustrates the idea of the existence of weak solutions to the model(P),that requires passing to the limits,Δt→0+and M→∞.The time stepΔt is subsequently linked to the cutoff parameter M>1 by positing a demand thatΔt=o(M−1),as M→∞,with the result that the cutoff parameter becomes the only parameter in the problem(PΔt M).The solutions can be bounded,independ-ent of M,with the use of special energy estimates,as demonstrated herein.Then,these M-independent bounds on the relative entropy are employed with the purpose of deriving M-independent bounds on the time-derivatives.Additionally,compactness arguments were utilised to explore the convergence of the finite element approximate problem.The conclu-sion was that a weak solution for(P)existed.Finally,we introduced the error estimate and the implicit scheme was used to perform simulations in one and two space dimensions.
文摘In this paper,a finite element scheme for the attraction-repulsion chemotaxis model is analyzed.We introduce a regularized problem of the truncated system.Then we obtain some a priori estimates of the regularized functions,independent of the regularization parameter,via deriving a well-defined entropy inequality of the regularized problem.Also,we propose a practical fully discrete finite element approximation of the regularized problem.Next,we use a fixed point theorem to show the existence of the approximate solutions.Moreover,a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived.In addition,the uniqueness of the fully discrete approximations is preformed.Finally,we discuss the convergence to the fully discrete problem.
