Generalized Finite Element Methods

1. Generalized Finite Element Methods Mathematical Preliminaries Suvranu De

2. Sets • Linear (Vector) Spaces • Inner product spaces and Cauchy Schwartz inequality • Normed spaces • Hilbert and Banach spaces • Some Function Spaces • Continuity spaces • Lebesgue spaces • Sobolev (Hilbert) spaces • Linear Operators • Linear functionals • Bilinear forms • Finite Dimensional Spaces

3. Sets

4. A set is a well defined collection of objects (e.g., scalars, vectors or functions) which are known as members or elements of the set Sets Examples of sets of scalars Subsets of the real line

5. Sets Examples of sets of vectors

6. Sets Examples of sets of functions Sets of functions are most important for understanding the theory of differential equations. The theory of a set of functions is developed very similar to the theory of a set of vectors.

7. Sets Cartesian products of sets: Given two sets A and B, their Cartesian product is a set C=AxB which contains all ordered pairs (x,y) where xA and yB 8 7 1 In general AxB≠ BxA 2 3

8. Sets: Supremum and infimum A upper bounds lower bounds • ‘A’ can have manyupper and lower bounds • If ‘A’ has bothan upper and a lower bound, it is said to be bounded

9. Sets: Supremum and infimum Note: A closed interval has a maximum and a minimum whereas an open interval does not have such maxima or minima However, a set is not that different from a set and it has a least upper bound which is the smallest of all its upper bounds

10. Sets: Supremum and infimum Note: A closed interval has a maximum and a minimum whereas an open interval does not have such maxima or minima However, a set is not all that different from a set and it has a least upper bound (supremum) which is the smallest of all its upper bounds and a greatest lower bound (infimum) which is the largest of all its lower bounds, even though these bounds do not belong to the set. It is therefore advisable to talk about inf and sup rather than max and min

11. Linear (Vector) SpacesNormed and inner product spaces

12. Linear (Vector) spaces A linear space (X) is a nonempty set such that A linear space is “closed” under addition and scalar multiplication e.g., (1) Vectors in the Euclidean n-space form a linear space addition is defined as

13. Linear spaces scalar multiplication is defined as hence

14. Linear spaces e.g., (2) the set of polynomials of degree at most 3 defined on [0,1] is a linear space e.g., (3) Let be a bounded domain in with boundary Is the following set a linear space?

15. An inner product space (X) is a linear space equipped with an inner product Inner product space/ pre-Hilbert space an inner product is a map that assigns a real number to every ordered pair and similar to a dot product in vector algebra satisfies the following properties: • If then x and y are said to be orthogonal

16. Aside: What does ‘iff’ mean? ‘if and only if’ Theorem proving: If we are given two mathematical statements A and B

17. Inner product spaces e.g., if the inner product (scalar product) satisfies

18. The Cauchy-Schwartz inequality is a property of an inner product space Inner product spaces proof: This is a quadratic in a whose graph either lies above the a axis or just touches it. Hence it should have either coincident roots or imaginary roots, i.e., the discriminant must be nonpositive

19. Normed linear space The normed linear space is one on which a norm is defined A norm satisfies the following properties

20. Norms Positive definiteness: in analogy with the “length” of a vector, a norm will always be positive. It will be 0 iff the original quantity is 0. Positive homogeneity: see figure to realize why ‘positive homogeneity’ and not simple homogeneity is necessary. Triangle inequality: see figure

21. Norms Norms are not unique. e.g., “max” / “sup” norm

22. Natural norm A norm is a primitive concept and does not require an inner product for its definition. However, in an inner product space the inner product induces a natural norm Proof (that the above inner product is a norm):

23. Natural norm Cauchy-Schwartz inequality

24. Natural norm Example (1) is an inner product space where the inner product induces a natural norm which represents the length of the vector Example (2) is a set of all polynomials of degree at most ‘m’ defined on the closed interval [0,1]. We define the L2 inner product Check that this satisfies symmetry, linearity and positive definiteness which induces a natural norm

25. Seminorm Semi-norm is just like a norm, except that it is positive semidefinite, i.e., e.g., is a seminorm since

26. Completeness: is a property whereby all Cauchy sequences converge to a member of X, i.e. Banach space: a normed linear space that is complete (it is not necessary for a Banach space to have an inner product) Hilbert space: an inner product space that is complete (i.e., Hilbert spaces are Banach spaces but not vice versa)

27. Some Function Spaces • Continuity Spaces • Lebesgue Spaces • Sobolev (Hilbert) Spaces

28. Continuity Spaces The continuity spaces are nested We define

29. Lebesgue Spaces Since many functions that are of practical interest in the solution of physical problems are not continuous (e.g., Heaviside functions) but integrable, instead of comparing functions in terms of their continuity, we compare functions in terms of their integrability. Lebesgue spaces are, in general, Banach spaces where the following norm is finite and These spaces are also nested

30. Lebesgue Spaces Special cases: space in which the functions are “square integrable” Unlike other Lp spaces, this is a Hilbert space and has the inner product which induces the norm

31. Lebesgue Spaces Special cases: Like other Lp spaces, this is not a Hilbert space (but a Banach space) and therefore does not have an inner product. The corresponding norm is known as the “sup” norm i.e., this is “essentially” the largest value of v on the domain

32. Sobolev (Hilbert) Spaces These are the most important function spaces for the numerical solution of partial differential equations. The Sobolev space contains functions that are square integrable (i.e., in ) and whose (weak) partial derivatives of order upto and including mare also square integrable where

33. Sobolev (Hilbert) Spaces Since these function spaces are Hilbert spaces, they have an inner product the norm induced by this inner product is Of course, Inclusion property:

34. Sobolev Embedding Theorem How are Sobolev spaces related to the continuity spaces? This theorem establishes the continuity of Sobolev spaces. ‘m’and ‘k’ are nonnegative integers and For continuity, k=0 e.g., d=1 (i.e. in 1D) For d=2 (i.e. in 2D)

35. Linear Operators • Linear Functionals • Bilinear forms

36. Operators An operator is a mapping from one space X (Domain space) to another space Y (Image space) Why bother? The differential equation Au=f may be viewed as an operation with A being the operator X (Domain) Y(Image)

37. Linear operators on Hilbert spaces In this class, we will concentrate on X and Y being Hilbert spaces (i.e, have norms and inner products) and the special case that the operator is linear. i.e. Or, in one statement

38. Linear Functionals (l) Linear functionals are special linear operators that map elements of X to real numbers e.g., is a linear functional using Cauchy Scwartz inequality hence this is also bounded The space of bounded linear functionals on X is known as the dual space of X i.e, X’

39. Bilinear form (a) Bilinear forms are special linear operators that map pairs of elements to real numbers and have the following properties Note: for an inner product space, the inner product is a bilinear form e.g., is a bilinear form is a bilinear form

40. Bilinear form (a) Symmetric Positive definite Continuity (boundedness) of a bilinear form H-ellipticity of a bilinear form Notice that a H-elliptic bilinear form must be positive definite – why?

41. Finite Dimensional Spaces

42. Finite Dimensional Spaces Every member of a finite dimensional space can be expressed as a linear combination of a (finite number of) selected subset of the members of that space Why bother? The solution of partial differential equations exist in infinite dimensional spaces, however, we always approximate them in finite dimensional spaces.

43. Span Span: A finite set {u1, u2, ...un} of elements of a vector space X is said to span X, written as X=span{u1, u2, ...un} if every may be written as a finite linear combination of the set, i.e. since I can express any polynomial of order at most in terms of these functions. But notice that the elements of the basis are not all independent.

44. Basis • A finite set {u1, u2, ...un} of elements of a vector space X is said to be the “basis” of X if • X=span{u1, u2, ...un} , and • the set {u1, u2, ...un} is linearly independent The set {1,x,x2,x3} is linearly independent. Hence, it is also the basis of P3[0,1]. NOTE: The basis is NOT unique. The following set is another basis: {1-x,1+x,x2,x3} The components of a polynomial p(x)=2x-x2+x3 wrt to the first basis are {0,2,-1,1} and wrt to the second basis is {-1,1-1,1} Dimension of basis:

45. Representation by matrices A linear operator from any arbitrary finite dimensional space to another can be represented as a matrix