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Institutionen för naturvetenskap och teknik Sobolev Spaces and the Finite Element Method Johan Davidsson Örebro universitet Institutionen för naturvetenskap och teknik Självständigt arbete för kandidatexamen i matematik, 15 hp Sobolev Spaces and the Finite Element Method Johan Davidsson May 2018 Supervisor: Farid Bozorgnia Examiner: Marcus Sundhäll Abstract In this essay we present the Sobolev spaces and some basic properties of them. The Sobolev spaces serve as a theoretical framework for studying solutions to partial differential equations. The finite element method is pre- sented which is a numerical method for solving partial differential equations. Contents 1 Preliminaries 5 1.1 Notation . .5 1.2 Measure, Integration and Almost Everywhere . .5 1.3 Functional Analysis . .9 1.4 Distributions . 10 2 Sobolev Spaces 11 2.1 Hölder Spaces . 11 2.2 Weak Derivative . 14 2.3 Definitions and Properties of Sobolev Spaces . 16 3 Finite Element Method 23 3.1 Weak Solution . 23 3.2 Finite Element Method . 24 4 Future Work 27 Chapter 1 Preliminaries In Section 1.1 we present some notations that will be used frequently through- out this text. Section 1.2 deals with some basic concepts in measure theory such as an extension of the length of intervals to open sets. We also present the concept that a relation can hold almost everywhere. For a full review on measure theory the reader can look up Folland [4]. With this it will make sense to consider solutions to partial differential equations satisfying a given equation almost everywhere instead of pointwise. In Section 1.3 we state some functional analysis and at the end we briefly review the concept of distributions. For a review on functional analysis and distributions one can read Debnath and Mikusiński [2]. 1.1 Notation n Let Ω denote an open subset of R and @Ω denote the boundary of Ω. The set N will denote the natural numbers including zero. The closure of a set Ω will be denoted by Ω. The derivative will be denoted by D. A multi-index α is an n-tuple α = (α1; : : : ; αn) where αi ≥ 0 for i = 1; : : : n. Let α and β be multi-indices, below we introduce some useful notations: Sum and difference: α ± β = (α1 ± β1; : : : ; αn ± βn): Sum of components: jαj = α1 + ::: + αn: α @α1 @αn Partial derivatives: D = α1 ::: αn : @x1 @xn 1.2 Measure, Integration and Almost Everywhere Definition 1.2.1. A σ-algebra is a nonempty family M containing subsets of a set X such that 1 S 1. If A1;A2;::: 2 M =) Ai 2 M: i=1 5 2. If A 2 M =) Ac 2 M: σ-algebras are used as domains for measure. A σ-algebra is intuitively a family of nice sets that we can assign a measure to. The elements of M are therefore called measurable sets. Definition 1.2.2. Let X be a set with σ-algebra M.A measure is a function µ : M ! [0; 1] such that 1. µ(;) = 0: 1 2. If fAig1 is a sequence of pairwise disjoint sets in M, the following must be true 1 1 [ X µ( Ai) = µ(Ai): i=1 i=1 This function µ relates a set with a number that can be interpreted as the size of that set. Definition 1.2.3. Let P(X) be the powerset of X. An outer measure is a function µ∗ : P(X) ! [0; 1] such that 1. µ∗(;) = 0. 2. µ∗(A) ≤ µ∗(B) if A ⊂ B. 1 1 ∗ S P ∗ 3. µ ( Ai) ≤ µ (Ai). i=1 i=1 Let µ be a measure defined on a σ-algebra M over X. Then the pair (X; M) is called a measuable space and the triplet (X; M; µ) is called a mea- sure space. Definition 1.2.4. Let A ⊂ R, then we define outer Lebesgue measure of a set A as ( 1 1 ) ∗ X [ m (A) = inf (bi − ai): A ⊂ [ai; bi] : i=1 i=1 Definition 1.2.5. The family of all Lebesgue measurable sets, L(R), consists of sets A such that the following holds for all E ⊂ R m∗(E) = m∗(E \ A) + m∗(E \ Ac); ∗ where m is defined in Definition 1.2.4. L(R) is a σ-algebra. Definition 1.2.6. Let A 2 L(R) then we define the Lebesgue measure m(A) as the outer measure of A, m(A) = m∗(A): 6 The Lebesgue measure generalizes the notion of length, area and volume n to arbitrary sets in higher dimension. When working with R one usually consider µ to be the Lebesgue measure, which is what we will do from now on. Given a measure space (Ω; M; µ), a null set is a set A ⊂ Ω such that µ(A) = 0. We state a formal definition below. Definition 1.2.7 (Null set). Let A ⊂ Ω. The set A is called a null set if and only if for every > 0 there is a countable family of n-cubes fAig that covers A such that 1 X µ(Ai) = . i=1 Example 1.2.1. Let c be a point in R. Let > 0 and choose Ii = (c − 2i+1 ; c + 2i+1 ). We have that 1 [ fcg ⊂ Ii i=1 and we get the following 1 X 1 1 2 2 µ(Ii) = + · + · ( ) + ::: = = . 2 2 2 2 2 1 − 1 i=1 2 We conclude that µ(c) = 0. Definition 1.2.8 (Almost everywhere). Let (X; M; µ) be a measure space. We say that a property holds almost everywhere, when it is true everywhere except for a set A ⊂ X such that µ(A) = 0. Example 1.2.2. Let f; g : R ! R, g(x) = x and ( x if x 6= 1; f(x) = 0 if x = 1: We can say that f = g almost everywhere since µ(1) = 0. Next we will develop the Lebesgue integral, we start off by defining a measurable function. Recall that any mapping f : X ! Y induces a mapping f −1 defined by f −1(A) = fx 2 X : f(x) 2 Ag: Definition 1.2.9. Let (X; M); (Y; N) be measure spaces. A mapping f : X ! Y is called measurable if f −1(A) 2 M for all A 2 N. Definition 1.2.10. Let A ⊂ X, the indicator function χA : X ! f0; 1g is defined as ( 1 if x 2 A; χA(x) = 0 if x2 = A. 7 First off we will present the simple functions, they are easy to construct an integral from. The simple functions has properties that will help us define an integral for more general functions. Definition 1.2.11. Let A1;:::;An be a sequence of disjoint measurable sets and a1; : : : ; an 2 R.A simple function f : X ! R is a function of the form n X f(x) = aiχAi (x): i=1 Pn Definition 1.2.12. For a positive simple function f = i=1 aiχAi we define the integral of f with respect to the measure µ as n Z X f dµ = aiµ(Ai): A i=1 So we have an integral for simple functions and from [4] we know that measurable functions can be approximated by simple functions. Definition 1.2.13. Let f be a positive measurable function. We define the integral of f as Z Z f dµ = sup φ dµ, φ≤f where the supremum is taken over all simple functions φ such that φ ≤ f. We developed the integral for positive measurable functions. For the general case you split a function f into its positive and negative parts, that is f = f + − f −. We say that a measurable function f is integrable if both R f + dµ and R f − dµ is finite. For details on this the reader can look in [4, chapter 2]. From the definition we see that if f; g are integrable and a; b 2 R. The following properties hold Z Z Z af + bg dµ = a f dµ + b g dµ, if f ≤ g we get that Z Z f dµ ≤ g dµ. Theorem 1.2.1. (The dominated convergence theorem) Let ffng be a se- quence of measurable functions such that fn ! f pointwise almost everywhere as n ! 1. Also let jfj ≤ g where g is integrable. Then we have that f is integrable and Z Z f = lim fn: n!1 8 1.3 Functional Analysis Definition 1.3.1 (Normed space). A vector space Y equipped with a norm is called a normed space. The norm generalizes the notion of length into abstract vector spaces. Definition 1.3.2 (Bilinear functional). A functional F (x; y) on a vector space E is called a bilinear functional if the following holds: 1. F (ax1 + bx2; y) = aF (x1; y) + bF (x2; y). 2. F (x; ay1 + by2) = conj(a)F (x; y1) + conj(b)F (x; y2). for any scalars a; b and x; x1; x2; y; y1; y2 2 Y . Where conj denote the com- plex conjugate. Definition 1.3.3 (Coercive functional). A bilinear functional F (x; y) on a normed space Y is called coercive if there exist a positive constant C such that F (x; x) ≥ Ckxk2 8x 2 Y: Definition 1.3.4 (Complete space). Let (Y; k·k) be a normed space. If every cauchy sequence in Y has a limit that is also in Y , then (Y; k·k) is called a complete space(or Banach space). Definition 1.3.5 (Inner product space). A vector space equipped with an inner product h·; ·i is called an inner product space.
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