| Delay partial differential equations(DPDEs)are widely used in economics,physics,ecology,biological systems,pharmacology,epidemiology,engineering control,computer aided design,nuclear engineering,climate models etc.,and have been getting more and more attention.In general,it's very difficult to express the exact solution of DPDEs with analytical expressions.Therefore,investigations on numerical methods are significant for solving this kind of problem.In this article,the research object is the convection-diffusion-reaction equation with time delay,which is important for DPDEs.This paper aims for studying the collocation finite element method,we discretize the PDE with classical collocation method in time and continuous finite element method in space.The whole investigation can be separated into four parts.Firstly,proposing the fully discrete scheme generated by discretizing the original problem with the two numerical methods as mentioned above in space and time,respectively;Secondly,giving the theoretical analysis of the numerical scheme and obtaining the convergence order;Thirdly,presenting a modified algorithm based on the original one(the fully discrete scheme)for solving the DPDE in this paper.The critical idea is transforming the original PDE into the equivalent dual PDE with a transformation,computing the dual approximation solution with the fully discrete scheme and transforming it into the approximation solution of original problem.What's more? There will be comparison between the original algorithm and the modified one,discussion about the advantages and disadvantages of the modified algorithm as well as the reason of presenting it in this paper.Finally,confirming the theoretical result along with the feasibility of the two algorithms with numerical experiments.At the end of this article,the research direction in the future will be proposed according to the emerging problem during the investigation process,which is related to the disadvantages of modified algorithm. |
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