High-precision Space-time Finite Volume Element Method For Parabolic Equations

Research on this paper to be carried out mainly from two parts.We solve a class of convection-diffusion equations by using the time-discontinuous space-time finite volume element method in the first part.In the second part,we solve a class of parabolic equations by applying the variable grid continuous space-time finite volume element method.In the first section,we combine the idea of the time-discontinuous space-time element with the method of super-convergent finite volume element based on cubic Lagrange inter-polation of equidistant nodes,where the derivative super-convergence points of the cubic Lagrange interpolation are taken as the dual split points.Moreover,we establish the time-discontinuous space-time finite volume element format of the convection-diffusion equation by introducing the interpolation projection operator.Combining the finite vol-ume element analysis with Lagrange interpolation polynomials fixed by Radau integral points,we prove the~?(~2)-norm optimal order error estimate of the approximate so-lutions.Finally,a numerical example is given to demonstrate the theoretical analysis results as well as the feasibility and effectiveness of the method.In the second section,we use the opinion of the variable grid space-time continuous element and the theory thought of the optimal stress points to construct a variable grid space-time continuous finite volume element scheme for parabolic equation.The finite volume element analysis is combined the Lagrange interpolation polynomials and the corresponding Gauss quadrature rules which are constructed by the Legendre and Lobatto points respectively.The existence and uniqueness of numerical solutions are proved in the condition that each time level can correspond to different space mesh structures,and the optimal error estimates of~?(~2)and~?(~1)-norm of the approximate solutions are given.Finally,a numerical example is presented to certify the validity and feasibility of the format.Furthermore,the rationality of the theoretical analysis results can be confirmed through numerical test.