Cubic Superconvergent Finite Volume Element Method For One-dimensional Elliptic And Parabolic Equations

As an important class of numerical method for solving differential equations, finite volume element method, which has the simplicity of finite difference and the accuracy of finite element, combines the advantages of the former two methods and has been widely used in computational fluid mechanics and so on. In this paper, we present cubic superconvergent finite volume element method for one dimensional elliptic and parabolic problems. Two chapters are included.In chapter one, the method based on cubic Lagrange interpolation associated with four equidistant nodes is presented.§1.1 is an introduction of finite volume element method and derivative superconvergent points of cubic Lagrange interpolation , i.e. for reference element [-1,1], if the cubic Lagrange interpolation is based on four equidistant nodes (-1,u(-1)), (-l/3,u(-l/3)), (1/3, u(l/3)), (l,u(l)), derivatives usually have third order accuracy. However, higher accuracy is obtained in {±(?)5/3,0}, which are derivative superconvergent points of cubic Lagrange interpolation.In§1.2, superconvergent finite volume element method based on cubic Lagrange interpolation for one-dimensional elliptic problem is considered. On each element [x_3i-3,x_3i, we choose cubic Lagrange interpolated function associated with the nodes (x_3i-3,u_3i-3,(x_3i-2,u_3i-2,(x_3i-1,x_3i-1, (x_3i,u_3i as trial function, piecewise constant function as test function, and derivative superconvergent points as dual partition nodes. Concrete computational scheme is derived and H¹ norm, L² norm error estimates are given, we prove that the scheme has optimal third order accuracy with respect to H¹ norm and fourth order accuracy with respect to L² norm. We also obtain that the scheme has fourth order superconvergence for derivatives at optimal stress points. Numerical examples verify the theoretical analysis and some of problems with singular sources are also provided.§1.3 generalizes the above ideas to one-dimensional parabolic problem, we obtain a scheme and the corresponding theoretical results with L² norm. Numerical experiment shows the efficiency of the method.In the second chapter, superconvergent finite volume element method based on cubic spline interpolation is presented.§2.1 introduces the applications of cubic spline interpolation in solving differential equations numerically, relevant knowledge is included and derivative superconvergent points are derived, i.e. for each element [x_i-1,x_i], interval midpoints are derivative superconvergent points for cubic spline interpolation. In§2.2, two point boundary value problems with various kinds of boundary conditions are studied. The convergence analysis with discrete H¹ norm and discrete L² norm is given and nonlinear problems and source term with singularity problem are discussed. In§2.3, finite volume element method base on cubic spline interpolation for one-dimensional parabolic problem is further studied, computational scheme is provided and error estimate is proved. Numerical examples are included to show the high efficiency and the adaptation of the method.