Refining the Galerkin method error estimation for parabolic type problem with a boundary condition

Abstract

The article considers a parabolic-type boundary value problem with a divergent principal part, when the boundary condition contains the time derivative of the required function: { ut−d/dxiai(x,t,u,∇u)+a(x,t,u,∇u)=0,a0ut+ai(x,t,u,∇u)cos(v,xi)=g(x,t,u,),(x,t)∈St, u(x,0)= u0(x), x∈Ω Such nonclassical problems with boundary conditions containing the time derivative of the desired function arise in the study of a number of applied problems, for example, when the surface of a body, whose temperature is the same at all its points, is washed off by a well-mixed liquid, or when a homogeneous isotropic body is placed in the inductor of an induction furnace and an electro-magnetic wave falls on its surface. Such problems have been little studied, therefore, the study of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant. In this paper, the definition of a generalized solution of the considered problem in the space H˜1,1(QT) is given. This problem is solved by the approximate Bubnov-Galerkin method. The coordinate system is chosen from the space H1(Ω). To determine the coefficients of the approximate solution, the parabolic problem is reduced to a system of ordinary differential equations. The aim of the study is to obtain conditions under which the estimate of the error of the approximate solution in the norm H1(Ω) has order O(hk−1) The paper first explores the auxiliary elliptic problem. When the condition of the ellipticity of the problem is satisfied, inequalities are proposed for the difference of the generalized solution of the considered parabolic problem with a divergent principal part, when the boundary condition contains the time derivative of the desired function and the solution of the auxiliary elliptic problem. Using these estimates, as well as under additional conditions for the coefficients and the function included in the problem under consideration, estimates of the error of the approximate solution of the Bubnov-Galerkin method in the norm H1(Ω) of order O(hk−1) for the considered nonclassical parabolic problem with divergent principal part, when the boundary condition contains the time derivative of the desired function.