University:
Massachusetts Institute of Technology
Course:
16.920J | Numerical Methods for Partial Differential Equations (SMA 5212)
Academic year:

2021
Views:

67

Pages:

30
Author:

Kontinenz5r96q

1 Motivation The Poisson problem has a strong formulation; a minimization formulation; and a weak formulation. The minimization/weak formulations are more general than the strong formulation in terms of regularity and admissible data. • The minimization/weak formulations are defined by: a space X; a bilinear form a; a linear form The minimization/weak formulations identify ESSENTIAL boundary conditions, Dirichlet reflected in X; NATURAL boundary conditions, Neumann reflected in a. f. The points of departure for the finite element method are: the weak formulation (more generally); or the minimization statement (if a is SPD). 2 The Dirichlet Problem 2.1 Strong Formulation Find u such that = / in 2 on Γ The boundary condition u = 0 is denoted "homogeneous Dirichlet." We consider Neumann boundary conditions (imposed) in Section 3, and inhomogenеома Dirichlet boundary conditions in Section 4. where 82 and is a domain in R³ with boundary Γ. 1 но (w,v upon multiplying by [v. We obtain strict equality if w is proportional to e Spaces (Ω), Η (Ω), Η (Ω) It is a simple matter to show that H(0) and H¹ (0) are inner product spaces. It is decidedly less simple to show that these spaces are complete, though they do indeed have this property, and hence are indeed Hilbert spaces. (We note that an alternative view of H() in fact defines this space as the completion of a class of infinitely smooth functions with respect to the H norm.) Completion (and relatedly, closure, which ensures that the limit of a sequence of members of a subset is itself a member of the subset) is a rather theoretical notion that we will simply take for granted; but it is important, making sure that we do not find ourselves in a situation in which the limit of a sequence of functions has very different properties than each member of the sequence. We will briefly discuss the latter again below. We can easily generalize the spaces integer m. We do so in R, say (1) to H (1) for any non-negative (0,1), to avoid multi-indices for mixed derivatives in higher dimensions. We thus have that H™ (Ω) = dr < 00, with associated inner product da < 00, ()<). dr < 00 (w)(0) = الدر dai dai dr. and norm -(()) Σ =(13)||| These spaces are important not only in understanding well-posedness of weak statements, but also in expressing the converge convergence rate of the finite element method. We shall have most need for H(2), H (2), and H²(Q); the latter re- quires that, in addition to the function and the derivative, the second derivative must also have finite energy. We also introduce the H™ (2) semi-norms an --(()) which includes only the mt derivative. The H¹ semi-norm is simply w 1/2 (w de) In In general, a semi-norm must satisfy all the properties of a norm 10 ezcept that it is permitted to vanish for w0. However, for the case of H(1), the H¹ semi-norm is equivalent to the full H¹ norm, by which we mean that (0)≤(0) Vw∈(0); the condition that we H(9) must vanish on I ensures that uw is not free to "float." The left-hand inequality is known as the Poincaré-Friedrichs inequality, and can also be related to the minimum eigenvalue of the Dirichlet Laplacian problem through the Rayleigh quotient. We shall prove this result for a partic ular problem in a later lecture. The Lebesgue spaces, LP(2) Another set of spaces that are very important in finite element analysis are the Lebesgue spaces, LP(N), p≥ 1, which are not Hilbert spaces except in the particular (and perhaps most important) case p 2: for p = 2, L(N) = H°(Ω), with inner product (w.)2(2) wudA and norm More generally, with norm 1/2 (0)=(44) ={01/10/dA < 0}, LP(9) = =(1/4): it can be shown by the Holder inequality that (0) CL() for q≥p. The sense of integration here is (appropriately enough) Lebesgue integration. We do not enter into the theory of integration here except to note that Lebesgue Integration is very forgiving of (very) occasional omissions the Lebesgue integral will not change if we change the value of the integrand only on a set of "zero measure" (e.g., a point, or a finite set of points, or a countably infinite set of points). We can see the practical import of the Lebesgue definition of integration by considering L (N), that is, L(2) as poo. It is clear that as p→ oo our norm will pick up only the largest value of vl, so we might be tempted to write [[supe where we recall that sup means we look for the least upper bound, that is, the smallest value C such that (r) < C for all zn. However the above would say that, for a function which is zero everywhere but equal to, say, 10 at one 11

Finite Element Methods for Elliptic Problems; Variational Formulation: The Poisson Problem

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock