A Full Collocation Method For A Class Of Parabolic Equations

The collocation method is a numerical method which searches for the approximate solu-tion of the operator function by satisfying pure interpolation condition, and it was invented in the 1970s. The collocation method is essentially involved in determining an approximate solution by piecewise polynomial satisfying the differential equation and boundary condition exactly at certain points. The collocation method needn't calculate numerical integration, and its approxi-mation equation is easy to form and it has simple calculation and high convergence, so it is widely used for solving both engineering and computing mathematics. Collocation points usually use the nodes of Gauss quadrature formula, and choose piecewise Hermite bicubics polynomials as the approximative space, convergence rate can reach h4. Spline collocation method at Gauss points is named orthogonal spline collocation method(OSC). Spline collocation method needn't compute numerical integral which increases the workload and effects the precision of coefficient matrix. As a result, the collocation method has easier implementation and higher convergence rate than the finite element method. From the above, collocation method is widely used in many fields, such as elliptic equations, parabolic equations and hyperbolic equations. But the most of using the collo-cation method for solving parabolic equations is in that way:collocation in space while difference in time. But collocation method has been scarcely used both in space and in time. In this paper, some researches and collocation scheme about the full collocation method will be given, optimal error estimate and the stability result will be also derived.This content is divided into three chapters.The first chapter gives the basic theories of the full collocation method. And it not only gives a partition of the space and time, let M1(r,δ) be the space of piecewise Hermite bicubics, which is the approximate space, and let M0(s,ε) be the time of piecewise Hermite bicubics, which is the approximate time, but also gives the collocation points. It introduces the full collocation scheme and the existence and uniqueness of its solution. In the end, it gives three important theorems about error estimate.The second chapter considers a full collocation method for a quasilincar parabolic system, and gives the collocation scheme and the existence and uniqueness of its solution.The third chapter considers a full collocation method for incompressible miscible displace- ment in porous media. It is of significance to study incompressible miscible displacement porous media for many engineering fields, such as the exploitation of the underground oil. Modern com-puting methods and techniques on the numerical simulation of fluid flow model is of great use in many fields, such as water injection and production. Description of the problem is the following coupled system:In this chapter, I use orthogonal collocation method to approximate the pressure equation, and use the full collocation method to approximate the concentration equation. It not only gives the full collocation scheme and the order of the solution, but also proves the existence and uniqueness of the solution. Optimal order estimates are derived for the errors in the approximate solutions. Finally, we find such a solution (P, C)∈Nh×M1(r,δ) satisfying: where τl is the Gauss quadrature point, and letτk0=tk. The order of the solution is the following:In the end of the chapter, it gives optimal order error estimate:Suppose (a)-(d), andΔt1=Δtn'+1, r≥3, s≥4 then we have the following optimal error estimate...