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My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin ?nite element methods as appliedtoparabolicpartialdi?erentialequations. Theemphasesandselection of topics re?ects my own involvement in the ?eld over the past 25 years, and my ambition has been to stress ideas and methods of analysis rather than to describe the most general and farreaching results possible. Since the formulation and analysis of Galerkin ?nite element methods for parabolic problems are generally based on ideas and results from the corresponding theory for stationary elliptic problems, such material is often included in the presentation. The basis of this work is my earlier text entitled Galerkin Finite Element Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, from 1984. This has been out of print for several years, and I have felt a need and been encouraged by colleagues and friends to publish an updated version. In doing so I have included most of the contents of the 14 chapters of the earlier work in an updated and revised form, and added four new chapters, on semigroup methods, on multistep schemes, on incomplete iterative solution of the linear algebraic systems at the time levels, and on semilinear equations. The old chapters on fully discrete methods have been reworked by ?rst treating the time discretization of an abstract di?erential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.

Inhaltsverzeichnis

Frontmatter

1. The Standard Galerkin Method

Abstract
In this introductory chapter we shall study the standard Galerkin finite element method for the approximate solution of the model initial-boundary value problem for the heat equation
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2. Methods Based on More General Approximations of the Elliptic Problem

Abstract
In our above discussion of finite element approximation of the parabolic problem, the discretization in space was based on using a family of finitedimensional spaces S h H10 = H10 (Ω), such that, for some r ≥ 2, the approximation property (1.10) holds. The most natural example of such a family in a plane domain Ω is to take for S h the continuous functions which reduce to polynomials of degree at most r−1 on the triangles τ of a triangulation T h of Ω of the type described in the beginning of Chapter 1, and which vanish on ∂Ω However, for r > 2 and in the case of a domain with smooth boundary, it is not possible, in general, to satisfy the homogeneous boundary conditions exactly for this choice. This difficulty occurs, of course, already for the elliptic problem, and several methods have been suggested to deal with it. In this chapter we shall consider, as a typical example, a method which was proposed by Nitsche for this purpose.
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3. Nonsmooth Data Error Estimates

Abstract
In this chapter we shall first discuss a smoothing property of the solution operator of a homogeneous parabolic equation which shows that the solution is regular for positive time even if the initial data are not. We shall then demonstrate that an analogous behavior for the finite element solution implies that optimal order convergence takes place for positive time even for nonsmooth initial data. We also show some other results which elucidate the relation between the convergence of the finite element solution and the regularity of the exact solution.
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4. More General Parabolic Equations

Abstract
In this chapter we shall briefly discuss the generalization of our previous error analysis to initial-boundary value problems for more general parabolic equations, in which we allow the elliptic operator to have coefficients depending on both x and t, to contain lower order terms, and to be nonselfadjoint and nonpositive. In order not to have to account for possible exponential growth of stability constants and error bounds we restrict our considerations to finite interval in time.
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5. Negative Norm Estimates and Superconvergence

Abstract
In this chapter we shall extend our earlier error estimates in L2 and H1 to estimates in norms of negative order. It will turn out that if the accuracy in L2 of the family of approximating spaces is O(h r ) with r > 2, then the error bounds in norms of negative order is of higher order than O(h r ). In certain situations these higher order bounds may be applied to show error estimates for various quantities of these higher orders, so called superconvergent order estimates. We shall exemplify this by showing how certain integrals of the solution of the parabolic problem, and, in one space dimension, the values of the solution at certain points may be calculated with high accuracy using the semidiscrete solution.
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6. Maximum-Norm Estimates and Analytic Semigroups

Abstract
The main purpose in this chapter is to discuss stability and smoothness estimates for the semidiscrete solution of the homogeneous heat equation with respect to the maximum-norm, and some consequences of such estimates for error bounds for problems with smooth and nonsmooth initial data. The semidiscrete solution is sought in a piecewise linear finite element space belonging to a quasiuniform family.
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7. Single Step Fully Discrete Schemes for the Homogeneous Equation

Abstract
In this chapter we consider single step fully discrete methods for the initial boundary value problem for the homogeneous heat equation, and show analogues of our previous stability and error estimates in the spatially semidiscrete case for both smooth and nonsmooth data. Our approach is to first study the discretization with respect to time of an abstract parabolic equation in a Hilbert space setting by using rational approximations of the exponential, which allows the standard Euler and Crank-Nicolson procedures as special cases, and then to apply the results obtained to the spatially discrete problem investigated in the preceding chapters. The analysis uses eigenfunction expansions related to the elliptic operator occurring in the parabolic equation, which we assume positive definite.
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8. Single Step Fully Discrete Schemes for the Inhomogeneous Equation

Abstract
In this chapter we shall continue our study of single step fully discrete methods and turn now to the approximation of the inhomogeneous heat equation. Following the approach of Chapter 7 we shall first consider discretization in time of an ordinary differential equation in a Hilbert space setting, and then apply our results to the spatially discrete equation. In view of the work in Chapter 7 for the homogeneous equation with given initial data, we now restrict ourselves to the case that the initial data vanish.
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9. Single Step Methods and Rational Approximations of Semigroups

Abstract
In this chapter we shall again study single step time stepping methods for a homogeneous parabolic equation in an abstract setting. This time we will use the semigroup approach and represent the time stepping operator as a Dunford-Taylor integral in the complex plane, which will allow us to treat more general elliptic operators than in the previous chapter. For the purpose of including also application to maximum-norm estimates with respect to a spatial variable, which will be given at the end of the chapter, the analysis will take place in a Banach space framework.
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10. Multistep Backward Difference Methods

Abstract
In this chapter we shall first consider approximations at equidistant time levels of parabolic equations in which the time derivate is replaced by a multistep backward difference quotient of maximum order consistent with the number of time steps involved. We show that when this order is at most 6, then the method is stable and has a smoothing property analogous to that of single step methods of type IV. We shall use these properties to derive both smooth and nonsmooth data error estimates. In the end of the chapter we shall also discuss the use of two-step backward difference operators with variable time steps.
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11. Incomplete Iterative Solution of the Algebraic Systems at the Time Levels

Abstract
In the fully discrete methods for the solution of parabolic equations which we have studied so far, a finite diminensional system of linear algebraic equations has to be solved at each time level of the time stepping procedure, and our analysis has always assumed that these systems are solved exactly. Because in applications these systems are of high dimension, direct methods are most often not appropriate, and iterative methods have to be used. Since the linear system to be solved at an individual time level is a discretization of an elliptic partial differential equation (with the step size occurring as a small parameter), methods normally used for elliptic problems are natural to apply here. In practice, only a moderate finite number of iterations can be carried out at each time level, and it thus becomes interesting to determine how many steps of the iterative algorithm are needed to guarantee that no loss occurs in the order of accuracy compared to the basic procedure in which the systems are solved exactly. For a successful iterative strategy it is also important to make a proper choice of the starting approximation at each time step.
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12. The Discontinuous Galerkin Time Stepping Method

Abstract
In the previous chapters we have considered fully discrete schemes for the heat equation which were derived by first discretizing in the space variables by means of a Galerkin finite element method, which results in a system of ordinary differential equations with respect to time, and then applying a finite difference type time stepping method to this system to define a fully discrete solution. In this chapter, we shall apply the Galerkin method also in the time variable and thus define and analyze a method which treats the time and space variables similarly. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q – 1, which is not necessarily continuous at the nodes of the defining partition.
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13. A Nonlinear Problem

Abstract
In this chapter we shall consider the application of our previous methods of analysis to a nonlinear model problem. For simplicity and concreteness, we restrict our attention to the situation in the beginning of Chapter 1, with a convex plane domain and with piecewise linear approximating functions. We also consider the problem on a finite interval J = (0, t in time; some of the constants in our estimates will depend on t, without explicit mention.
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14. Semilinear Parabolic Equations

Abstract
In the last chapter we considered discretization in both space and time of a model nonlinear parabolic equation. The discretization with respect to space was done by piecewise linear finite elements and in time we applied the backward Euler and Crank-Nicolson methods. In this chapter we shall restrict the consideration to the case when only the forcing term is nonlinear, but discuss more general approximations in the spatial variable. We shall begin with the spatially semidiscrete problem and first briefly study global conditions on the forcing term and the finite element spaces under which optimal order error estimates can be derived for smooth data, uniformly down to t = 0, and then turn our attention to the analysis for nonsmooth initial data. We then discuss discretization in time by the backward Euler method, in particular with reference to nonsmooth initial data.
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15. The Method of Lumped Masses

Abstract
In this chapter we shall consider a modification of the standard Galerkin method using piecewise linear trial functions, the so-called method of lumped masses. In this method the mass matrix is replaced by a diagonal matrix with the row sums of the original mass matrix as its diagonal elements. This can also be interpreted as using a quadrature rule for the corresponding L2 inner product.
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16. The H1 and H−1 Methods

Abstract
In this chapter we briefly discuss some alternatives to the Galerkin methods considered above which use other inner products than that in L2(Ω) to formulate the discrete problem. For simplicity we shall content ourselves with describing the situation in the case of a simple selfadjoint parabolic equation in one space dimension, and only study spatially semidiscrete methods.
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17. A Mixed Method

Abstract
In this chapter we shall consider a finite element method for our model parabolic equation which is based on a mixed formulation of the problem. In this formulation the gradient of the solution is introduced as a separate dependent variable, the approximation of which is sought in a different finite element space than the solution itself. One advantage of this procedure is that the gradient of the solution may be approximated to the same order of accuracy as the solution itself.
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18. A Singular Problem

Abstract
In this chapter we shall study the numerical solution of a singular parabolic equation in one space dimension which arises after reduction by polar coordinates of a radially symmetric parabolic equation in three space dimensions. We shall analyze and compare finite element discretizations based on two different variational formulations.
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19. Problems in Polygonal Domains

Abstract
In earlier parts of this book we have generally assumed the spatial domain Ω to have a smooth boundary ∂Ω, which has made it possible to guarantee that the solution of the initial-boundary value problem is sufficiently regular for the purpose at hand, provided the data of the problem are sufficiently smooth and satisfy certain compatibility conditions at t = 0. In this chapter we shall consider the case when Ω is a plane polygonal domain. In this case singularities will in general appear in the solution even for smooth compatible data, and this will affect the convergence properties of the approximating finite element solution. We shall analyze in some detail the case of piecewise linear finite elements. In this case, no special difficulties arise whenΩ is convex, but when Ω is nonconvex the singularities will normally reduce the rate of convergence both for elliptic and for parabolic problems.
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20. Time Discretization by Laplace Transformation and Quadrature

Abstract
In this chapter we consider an alternative to time stepping for the discretization in time of an initial value problem for a parabolic equation. We now use a representation of the solution as an integral along a smooth curve extending into the complex right half plane, with an integrand containing the resolvent of the associated elliptic operator. This integral is then evaluated to high accuracy by a quadrature rule. In this way the problem is reduced to a finite set of elliptic equations, which may be solved in parallel. The procedure is combined with finite element discretization in the spatial variables.
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Backmatter

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