Finite Element Method For One-dimensional Nonlocal Initial Boundary Value Problem

In this paper, we pay main attention to the equations of nonlocal initialboundary value problem. In Chapter 1, we simply introduce the application andcurrent researches of nonlocal problem , we also introduce the problem which isto be studied . In Chapter 2, we list out some basic knowledge we need. Inchapter 3, the elliptic equations are solved by finite element method. We firstdiscuss the homogeneous boundary elliptic equation (notice: the word'homoge-neous'is not the same as normal), for this aim, we construct a H_*¹ space which isa self-contained subspace of H¹, and defne an operator from H_*¹ to H₀¹. We alsoprove that ||u||₁ is equal to |u|₁ and |Pu|₁ is equal to |u|1 for random u∈H_*¹if At last, we set up a fnite element space,and get the optimal error estimate of L2. After that, the nonlocal homogeneousboundary value problems in a general way are studied( the word'homogeneous'isthe same as above), we prove there exits only one weak solution or fnite elementsolution if To prove our methods areefcient, we introduce two numerical examples. Chapter 4 lists out two discreteways for homogeneous nonlocal parabolic equation and nonhomogeneous nonlo-cal parabolic equation should be changed to a homogeneous one, after that, wegive a numeric example. In Chapter 5, we discuss the source parabolic equation,and then we provide a Back-Euler-Galerkin method for homogeneous and nonho-mogeneous source parabolic equation. To prove our methods are valid, we givetwo numerical examples.