Scientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
|
|
- Cora Nash
- 5 years ago
- Views:
Transcription
1 Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann Lecture 15 Slide 1
2 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient assumes smoothness of coefficients and at least second derivatives for the solution. δ may not be continuous what is then (δ u)? Approximation of solution u e.g. by piecewise linear functions what does u mean? Spaces of twice, and even once continuously differentiable functions is not well suited: Favorable approximation functions (e.g. piecewise linear ones) are not contained Though they can be equipped with norms ( Banach spaces) they have no scalar product no Hilbert spaces Not complete: Cauchy sequences of functions may not converge to elements in these spaces Lecture 14 Slide 20
3 Lecture 15 Slide 3 Cauchy sequences of functions Let be a Lipschitz domain, let V be a metric space of functions f : R Regard sequences of functions f n = {f n } n=1 V A Cauchy sequence is a sequence f n of functions where the norm of the difference between two elements can be made arbitrarily small by increasing the element numbers: ε > 0 n 0 N : m, n > n 0, f n f m < ε All convergent sequences of functions are Cauchy sequences A metric space V is complete if all Cauchy sequences f n of its elements have a limit f = lim n f n V within this space Lecture 14 Slide 21
4 Lecture 15 Slide 4 Completion of a metric space Let V be a metric space. Its completion is the space V consisting of all elements of V and all possible limits of Cauchy sequences of elements of V. This procedure allows to carry over definitions which are applicable only to elements of V to more general ones Example: step function 1, x ɛ ( x ɛ f ɛ (x) = ɛ )2 + 1, 0 x < ɛ ( x+ɛ ɛ )2 1, ɛ x < 0 1, x < ɛ ɛ 0 f (x) = { 1, x 0 1, else Lecture 14 Slide 22
5 Lecture 15 Slide 5 Riemann integral Lebesgue integral Let be a Lipschitz domain, let C c () be the set of continuous functions f : R with compact support. ( they vanish on ) For these functions, the Riemann integral f (x)dx is well defined, and f L 1 := f (x) dx provides a norm, and induces a metric. Let L 1 () be the completion of C c () with respect to the metric defined by the norm L 1. That means that L 1 () consists of all elements of C c (), and of all limites of Cauchy sequences of elements of C c (). Such functions are called measurable. For any measurable f = lim f n L 1 () with f n C c (), define the n Lebesque integral f (x) dx := lim f n (x) dx n as the limit of a sequence of Riemann integrals of continuous functions Lecture 14 Slide 23
6 Lecture 15 Slide 6 Examples for Lebesgue integrable (measurable) functions Bounded functions which are continuous except in a finite number of points Step functions Equality of L 1 functions is elusive as they are not necessarily continuous: best what we can say is that they are equal almost everywhere. In particular, L 1 functions whose values differ in a finite number of points are equal almost everywhere. Lecture 14 Slide 24
7 Lecture 15 Slide 7 Spaces of integrable functions For 1 p, let L p () be the space of measureable functions such that equipped with the norm f (x) p dx < ( f p = f (x) p dx These spaces are Banach spaces, i.e. complete, normed vector spaces. The space L 2 () is a Hilbert space, i.e. a Banach space equipped with a scalar product (, ) whose norm is induced by that scalar product, i.e. u = (u, u). The scalar product in L 2 is ) 1 p (f, g) = f (x)g(x)dx. Lecture 14 Slide 25
8 Lecture 15 Slide 8 Green s theorem for smooth functions Theorem Let u, v C 1 () (continuously differentiable). Then for n = (n 1... n d ) being the outward normal to, u i v dx = uvn i ds v i u dx Corollaries Let u = (u 1... u d ). Then ( d u i i v) dx = i=1 u v dx = v d (u i n i ) ds i=1 vu n ds v d ( i u i ) dx i=1 v u dx If v = 0 on : u i v dx = v i u dx u v dx = v u dx Lecture 14 Slide 26
9 Lecture 15 Slide 9 Weak derivative Let L 1 loc () be the set of functions which are Lebesgue integrable on every compact subset K. Let C0 () be the set of functions infinitely differentiable with zero values on the boundary. For u L 1 loc () we define iu by and α u by v i udx = u i vdx v C0 () v α udx = ( 1) α u i vdx v C 0 () if these integrals exist. For smooth functions, weak derivatives coincide with with the usual derivative Lecture 14 Slide 27
10 Lecture 15 Slide 10 Sobolev spaces For k 0 and 1 p <, the Sobolev space W k,p () is the space functions where all up to the k-th derivatives are in L p : W k,p () = {u L p () : α u L p () α k} with then norm u W k,p () = α u p L p () α k Alternatively, they can be defined as the completion of C in the norm u W k,p () 1 p W k,p 0 () is the completion of C 0 in the norm u W k,p () The Sobolev spaces are Banach spaces. Lecture 14 Slide 28
11 Lecture 15 Slide 11 Sobolev spaces of square integrable functions H k () = W k,2 () with the scalar product (u, v) H k () = α u α v dx α k is a Hilbert space. k,2 () = W0 () with the scalar product (u, v) H k 0 () = α u α v dx H k 0 α =k is a Hilbert space as well. For this course the most important: L 2 (), scalar product (u, v) L 2 () = (u, v) 0, = uv dx Inequalities: H 1 (), scalar product (u, v) H 1 () = (u, v) 1, = (uv + u v) dx H 1 0 (), scalar product (u, v) H 1 0 () = ( u v) dx (u, v) 2 (u, u)(v, v) u + v u + v Cauchy-Schwarz Triangle inequality Lecture 14 Slide 29
12 Lecture 15 Slide 12 A trace theorem The notion of function values on the boundary initially is only well defined for continouos functions. So we need an extension of this notion to functions from Sobolev spaces. Theorem: Let be a bounded Lipschitz domain. Then there exists a bounded linear mapping such that tr : H 1 () L 2 ( ) (i) c > 0 such that tr u 0, c u 1, (ii) u C 1 ( ), tr u = u Lecture 14 Slide 30
13 Lecture 15 Slide 13 Derivation of weak formulation Sobolev space theory provides a convenient framework to formulate existence, uniqueness and approximations of solutions of PDEs. Stationary heat conduction equation with homogeneous Dirichlet boundary conditions: λ u(x) = f (x) in u = 0 on Multiply and integrate with an arbitrary test function v C0 apply Green s theorem using v = 0 on ( λ u)v dx = fv dx λ u v dx = fv dx () and Lecture 14 Slide 31
14 Lecture 15 Slide 14 Weak formulation of homogeneous Dirichlet problem Search u H0 1 () (here, tr u = 0) such that λ u v dx = fv dx v H0 1 () Then, a(u, v) := λ u v dx is a self-adjoint bilinear form defined on the Hilbert space H0 1(). It is bounded due to Cauchy-Schwarz: a(u, v) = λ u v dx u H 1 0 () v H 1 0 () f (v) = fv dx is a linear functional on H1 0 (). For Hilbert spaces V the dual space V (the space of linear functionals) can be identified with the space itself. Lecture 14 Slide 32
15 Lecture 15 Slide 15 The Lax-Milgram lemma Theorem: Let V be a Hilbert space. Let a : V V R be a self-adjoint bilinear form, and f a linear functional on V. Assume a is coercive, i.e. α > 0 : u V, a(u, u) α u 2 V. Then the problem: find u V such that a(u, v) = f (v) v V admits one and only one solution with an a priori estimate u V 1 α f V Lecture 14 Slide 33
16 Lecture 15 Slide 16 Coercivity of weak formulation Theorem: Assume λ > 0. Then the weak formulation of the heat conduction problem: search u H0 1 () such that has an unique solution. Proof: a(u, v) is cocercive: λ u v dx = fv dx v H0 1 () a(u, v) = λ u u dx = λ u 2 H 1 0 () Lecture 14 Slide 34
17 Lecture 15 Slide 17 Weak formulation of inhomogeneous Dirichlet problem λ u = f in u = g on If g is smooth enough, there exists a lifting u g H 1 () such that u g = g. Then, we can re-formulate: λ (u u g ) = f + λ u g in u u g = 0 on Search u H 1 () such that u = u g + φ λ φ v dx = fv dx + λ u g v v H 1 0 () Here, necessarily, φ H0 1 () and we can apply the theory for the homogeneous Dirichlet problem. Lecture 14 Slide 35
18 Lecture 15 Slide 18 Weak formulation of Robin problem λ u = f in λ u n + α(u g) = 0 on Multiply and integrate with an arbitrary test function from C c (): ( λ u)v dx = λ u v dx + (λ u n)vds = λ u v dx + αuv ds = fv dx fv dx fv dx + αgv ds Lecture 14 Slide 36
19 Lecture 15 Slide 19 Weak formulation of Robin problem II Let a R (u, v) := f R (v) := Search u H 1 () such that λ u v dx + αuv ds fv dx + αgv ds a R (u, v) = f R (v) v H 1 () If λ > 0 and α > 0 then a R (u, v) is cocercive. Lecture 14 Slide 37
20 Lecture 15 Slide 20 Neumann boundary conditions Homogeneous Neumann: λ u n = 0 on Inhomogeneous Neumann: λ u n = g on Weak formulation: Search u H 1 () such that u v dx = gv ds v H 1 () Not coercive due to the fact that we can add an arbitrary constant to u and a(u, u) stays the same! Lecture 14 Slide 38
21 Lecture 15 Slide 21 Further discussion on boundary conditions Mixed boundary conditions: One can have differerent boundary conditions on different parts of the boundary. In particular, if Dirichlet or Robin boundary conditions are applied on at least a part of the boundary of measure larger than zero, the binlinear form becomes coercive. Natural boundary conditions: Robin, Neumann These are imposed in a natural way in the weak formulation Essential boundary conditions: Dirichlet Explicitely imposed on the function space Coefficients λ, α... can be functions from Sobolev spaces as long as they do not change integrability of terms in the bilinear forms Lecture 14 Slide 39
22 Lecture 15 Slide 22 The Galerkin method I Weak formulations live in Hilbert spaces which essentially are infinite dimensional For computer representations we need finite dimensional approximations The Galerkin method and its modifications provide a general scheme for the derivation of finite dimensional appoximations Finite dimensional subspaces of Hilbert spaces are the spans of a set of basis functions, and are Hilbert spaces as well e.g. the Lax-Milgram lemma is valid there as well Lecture 14 Slide 42
23 Lecture 15 Slide 23 The Galerkin method II Let V be a Hilbert space. Let a : V V R be a self-adjoint bilinear form, and f a linear functional on V. Assume a is coercive with coercivity constant α, and continuity constant γ. Continuous problem: search u V such that a(u, v) = f (v) v V Let V h V be a finite dimensional subspace of V Discrete problem Galerkin approximation: Search u h V h such that a(u h, v h ) = f (v h ) v h V h By Lax-Milgram, this problem has a unique solution as well. Lecture 14 Slide 43
24 Lecture 15 Slide 24 Céa s lemma What is the connection between u and u h? Let v h V h be arbitrary. Then α u u h 2 a(u u h, u u h ) (Coercivity) = a(u u h, u v h ) + a(u u h, v h u h ) = a(u u h, u v h ) (Galerkin Orthogonality) γ u u h u v h (Boundedness) As a result u u h γ α inf v h V h u v h Up to a constant, the error of the Galerkin approximation is the error of the best approximation of the solution in the subspace V h. Lecture 14 Slide 44
25 Lecture 15 Slide 25 From the Galerkin method to the matrix equation Let φ 1... φ n be a set of basis functions of V h. Then, we have the representation u h = n j=1 u jφ j In order to search u h V h such that it is actually sufficient to require a(u h, v h ) = f (v h ) v h V h a(u h, φ i = f (φ i ) (i = 1... n) ( n ) a u j φ j, φ i = f (φ i ) (i = 1... n) j=1 n a(φ j, φ i )u j = f (φ i ) (i = 1... n) j=1 AU = F with A = (a ij ), a ij = a(φ i, φ j ), F = (f i ), f i = F (φ i ), U = (u i ). Matrix dimension is n n. Matrix sparsity? Lecture 14 Slide 45
26 Lecture 15 Slide 26 Obtaining a finite dimensional subspace Let = (a, b) R 1 Let a(u, v) = λ(x) u vdx. Analysis I provides a finite dimensional subspace: the space of sin/cos functions up to a certain frequency spectral method Ansatz functions have global support full n n matrix OTOH: rather fast convergence for smooth data Generalization to higher dimensions possible Big problem in irregular domains: we need the eigenfunction basis of some operator... Spectral methods are successful in cases where one has regular geometry structures and smooth/constant coefficients e.g. Spectral Einstein Code Lecture 14 Slide 46
27 Definition of a Finite Element (Ciarlet) Triplet {K, P, Σ} where K R d : compact, connected Lipschitz domain with non-empty interior P: finite dimensional vector space of functions p : K R Σ = {σ 1... σ s } L(P, R): set of linear forms defined on P called local degrees of freedom such that the mapping Λ Σ : P R s p (σ 1 (p)... σ s (p)) is bijective, i.e. Σ is a basis of L(P, R). Lecture 15 Slide 27
28 Local shape functions Due to bijectivity of Λ Σ, for any finite element {K, P, Σ}, there exists a basis {θ 1... θ s } P such that σ i (θ j ) = δ ij (1 i, j s) Elements of such a basis are called local shape functions Lecture 15 Slide 28
29 Unisolvence Bijectivity of Λ Σ is equivalent to the condition (α 1... α s ) R s!p P such that σ i (p) = α i (1 i s) i.e. for any given tuple of values a = (α 1... α s ) there is a unique polynomial p P such that Λ Σ (p) = a. Equivalent to unisolvence: { dim P = Σ = s p P : σ i (p) = 0 (i = 1... s) p = 0 Lecture 15 Slide 29
30 Lagrange finite elements A finite element {K, P, Σ} is called Lagrange finite element (or nodal finite element) if there exist a set of points {a 1... a s } K such that σ i (p) = p(a i ) 1 i s {a 1... a s }: nodes of the finite element nodal basis: {θ 1... θ s } P such that θ j (a i ) = δ ij (1 i, j s) Lecture 15 Slide 30
31 Local interpolation operator Let {K, P, Σ} be a finite element with shape function bases {θ 1... θ s }. Let V (K) be a normed vector space of functions v : K R such that P V (K) The linear forms in Σ can be extended to be defined on V (K) local interpolation operator I K : V (K) P s v σ i (v)θ i P is invariant under the action of I K, i.e. p P, I K (p) = p: s Let p = αjθj Then, j=1 i=1 I K (p) = = s s s σ i(p)θ i = α jσ i(θ j)θ i i=1 s i=1 s α jδ ijθ i = j=1 i=1 j=1 j=1 s α jθ j Lecture 15 Slide 31
32 Local Lagrange interpolation operator Let V (K) = (C 0 (K)) I K : V (K) P v I K v = s v(a i )θ i i=1 Lecture 15 Slide 32
33 Simplices Let {a 0... a d } R d such that the d vectors a 1 a 0... a d a 0 are linearly independent. Then the convex hull K of a 0... a d is called simplex, and a 0... a d are called vertices of the simplex. Unit simplex: a 0 = (0...0), a 1 = (0, )... a d = (0... 0, 1). K = { x R d : x i 0 (i = 1... d) and } d x i 1 i=1 A general simplex can be defined as an image of the unit simplex under some affine transformation F i : face of K opposite to a i n i : outward normal to F i Lecture 15 Slide 33
34 Barycentric coordinates Let K be a simplex. Functions λ i (i = 0... d): λ i : R d R x λ i (x) = 1 (x a i) n i (a j a i ) n i where a j is any vertex of K situated in F i. For x K, one has 1 (x a i) n i = (a j a i ) n i (x a i ) n i (a j a i ) n i (a j a i ) n i = (a j x) n i = dist(x, F i) (a j a i ) n i dist(a i, F i ) = dist(x, F i) F i /d dist(a i, F i ) F i /d = dist(x, F i) F i K i.e. λ i (x) is the ratio of the volume of the simplex K i (x) made up of x and the vertices of F i to the volume of K. Lecture 15 Slide 34
35 Barycentric coordinates II λ i (a j ) = δ ij λ i (x) = 0 x F i d i=0 λ i(x) = 1 x R d (just sum up the volumes) d i=0 λ i(x)(x a i ) = 0 x R d (due to λ i (x)x = x and λ i a i = x as the vector of linear coordinate functions) Unit simplex: λ0(x) = 1 d i=1 xi λi(x) = x i for 1 i d Lecture 15 Slide 35
36 Polynomial space P k Space of polynomials in x 1... x d of total degree k with real coefficients α i1...i d : Dimension: P k = p(x) = 0 i 1...i d k i 1+ +i d k α i1...i d x i x i d d dim P k = ( ) d + k = k k + 1, d = (k + 1)(k + 2), d = 2 (k + 1)(k + 2)(k + 3), d = dim P 1 = d + 1 3, d = 1 dim P 2 = 6, d = 2 10, d = 3 Lecture 15 Slide 36
37 P k simplex finite elements K: simplex spanned by a 0... a d in R d P = P k, such that s = dim P k For 0 i 0... i d k, i i d = k, let the set of nodes be defined by the points a i1...i d ;k with barycentric coordinates ( i0 k... i d k ). Define Σ by σ i1...i d ;k(p) = p(a i1...i d ;k). Lecture 15 Slide 37
38 P 1 simplex finite elements K: simplex spanned by a 0... a d in R d P = P 1, such that s = d + 1 Nodes vertices Basis functions barycentric coordinates Lecture 15 Slide 38
39 P 2 simplex finite elements K: simplex spanned by a 0... a d in R d P = P 2, Nodes vertices + edge midpoints Basis functions: λ i (2λ i 1),(0 i d); 4λ i λ j, (0 i < j d) ( edge bubbles ) Lecture 15 Slide 39
40 General finite elements Simplicial finite elements can be defined on triangulations of polygonal domains. During the course we will stick to this case. For vector PDEs, one can define finite elements for vector valued functions A curved domain may be approximated by a polygonal domain h which is then triangulated. During the course, we will ignore this difference. As we have seen, more general elements are possible: cuboids, but also prismatic elements etc. Curved geometries are possible. Isoparametric finite elements use the polynomial space to define a mapping of some polyghedral reference element to an element with curved boundary Lecture 15 Slide 40
41 Conformal triangulations Let T h be a subdivision of the polygonal domain R d into non-intersecting compact simplices K m, m = 1... n e : = n e m=1 Each simplex can be seen as the image of a affine transformation of a reference (e.g. unit) simplex K: K m K m = T m ( K) We assume that it is conformal, i.e. if K m, K n have a d 1 dimensional intersection F = K m K n, then there is a face F of K and renumberings of the vertices of K n, K m such that F = T m ( F ) = T n ( F ) and T m F = T n F Lecture 15 Slide 41
42 Conformal triangulations II d = 1 : Each intersection F = K m K n is either empty or a common vertex d = 2 : Each intersection F = K m K n is either empty or a common vertex or a common edge d = 3 : Each intersection F = K m K n is either empty or a common vertex or a common edge or a common face Delaunay triangulations are conformal Lecture 15 Slide 42
43 Global interpolation operator I h Let {K, P K, Σ K } K Th be a triangulation of. Domain: D(I h ) = {v (L 1 ()) such that K T h, v K V (K)} For all v D(I h ), define I h v via s I h v K = I K (v K ) = σ K,i (v K )θ K,i K T h, i=1 Assuming θ K,i = 0 outside of K, one can write I h v = s σ K,i (v K )θ K,i, K T h i=1 mapping D(I h ) to the approximation space W h = {v h (L 1 ()) such that K T h, v h K P K } Lecture 15 Slide 43
44 H 1 -Conformal approximation using Lagrangian finite elemenents Conformal subspace of W h with zero jumps at element faces: V h = {v h W h : n, m, K m K n 0 (v h Km ) Km K n = (v h Kn ) Km K n } Then: V h H 1 (). Lecture 15 Slide 44
45 Zero jump at interfaces with Lagrangian finite elements Assume geometrically conformal mesh Assume all faces of K have the same number of nodes s For any face F = K 1 K 2 there are renumberings of the nodes of K 1 and K 2 such that for i = 1... s, a K1,i = a K2,i Then, v h K1 and v h K2 match at the interface K 1 K 2 if and only if they match at the common nodes v h K1 (a K1,i) = v h K2 (a K2,i) (i = 1... s ) Lecture 15 Slide 45
46 Global degrees of freedom Let {a 1... a N } = {a K,1... a K,s } K T h Degree of freedom map j : T h {1... s} {1... N} (K, m) j(k, m) the global degree of freedom number Global shape functions φ 1,..., φ N W h defined by φ i K (a K,m ) = { δ mn if n {1... s} : j(k, n) = i 0 otherwise Global degrees of freedom γ 1,..., γ N : V h R defined by γ i (v h ) = v h (a i ) Lecture 15 Slide 46
47 Lagrange finite element basis {φ 1,..., φ N } is a basis of V h, and γ 1... γ N is a basis of L(V h, R). Proof: {φ 1,..., φ N } are linearly independent: if N j=1 α jφ j = 0 then evaluation at a 1... a N yields that α 1... α N = 0. Let v h V h. It is single valued in a 1... a N. Let w h = N j=1 v h(a j )φ j. Then for all K T h, v h K and w h K coincide in the local nodes a K,1... a K,2, and by unisolvence, v h K = w h K. Lecture 15 Slide 47
48 Finite element approximation space P k c,h = Pk h = {v h C 0 ( h ) : K T h, v k T K P k } c for continuity across mesh interfaces. There are also discontinuous FEM spaces which we do not consider here. d k N = dim P k h 1 1 N v 1 2 N v + N el 1 3 N v + 2N el 2 1 N v 2 2 N v + N ed 2 3 N v + 2N ed + N el 3 1 N v 3 2 N v + N ed 3 3 N v + 2N ed + N f Lecture 15 Slide 48
49 P 1 global shape functions Lecture 15 Slide 49
50 P 2 global shape functions Node based Edge based Lecture 15 Slide 50
51 Global Lagrange interpolation operator Let V h = P k h I h : C 0 ( h ) V h N v v(a i )φ i i=1 Lecture 15 Slide 51
52 Quadrature rules Quadrature rule: K g(x) dx K l q l=1 ω l g(ξ l ) ξ l : nodes, Gauss points ω l : weights The largest number k such that the quadrature is exact for polynomials of order k is called order k q of the quadrature rule, i.e. k k q, p P k Error estimate: φ C kq+1 (K), 1 K K K p(x) dx = K φ(x) dx l q l=1 ch kq+1 K l q l=1 ω l g(ξ l ) ω l p(ξ l ) sup x K, α =k q+1 α φ(x) Lecture 15 Slide 52
53 Some common quadrature rules Nodes are characterized by the barycentric coordinates d k q l q Nodes Weights ( 1, 1 ) (1, 0), (0, 1), ( 1 + 3, 1 3 ), ( 1 3, ) 1, ( 1, ), ( 1 + 3, 1 3 ), ( 1 3, ) 8, 5, ( 1, 1, 1 ) (1, 0, 0), (0, 1, 0), (0, 0, 1), 1, ( 1, 1, 0), ( 1, 0, 1 ), (0, 1, 1 ) 1, 1, ( 1, 1, 1 ), ( 1, 1, 3 ), ( 1, 3, 1 ), ( 3, 1, 1 ), 9, 25, 25, ( 1, 1, 1, 1 ) (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) 2 4 ( 5 5, 5 5, 5 5, ) , 1, 1, , 1, 1, Lecture 15 Slide 53
54 Weak formulation of homogeneous Dirichlet problem Search u V = H0 1 () such that κ u v dx = fv dx v H0 1 () Then, a(u, v) := κ u v dx is a self-adjoint bilinear form defined on the Hilbert space H 1 0 (). Lecture 15 Slide 54
55 Galerkin ansatz Let V h V be a finite dimensional subspace of V Discrete problem Galerkin approximation: Search u h V h such that a(u h, v h ) = f (v h ) v h V h E.g. V h is the space of P1 Lagrange finite element approximations Lecture 15 Slide 55
56 Stiffness matrix for Laplace operator for P1 FEM Element-wise calculation: a ij = a(φ i, φ j ) = φ i φ j dx = Standard assembly loop: for i, j = 1... N do set a ij = 0 end for K T h do for m,n=0... d do s mn = λ m λ n dx end end K a jdof (K,m),j dof (K,n) = a jdof (K,m),j dof (K,n) + s mn K T h φ i K φ j K dx Local stiffness matrix: S K = (s K;m,n ) = λ m λ n dx K Lecture 15 Slide 56
Scientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1
Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,
More informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationChapter 3 Conforming Finite Element Methods 3.1 Foundations Ritz-Galerkin Method
Chapter 3 Conforming Finite Element Methods 3.1 Foundations 3.1.1 Ritz-Galerkin Method Let V be a Hilbert space, a(, ) : V V lr a bounded, V-elliptic bilinear form and l : V lr a bounded linear functional.
More informationLECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,
LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation
More informationChapter 5 A priori error estimates for nonconforming finite element approximations 5.1 Strang s first lemma
Chapter 5 A priori error estimates for nonconforming finite element approximations 51 Strang s first lemma We consider the variational equation (51 a(u, v = l(v, v V H 1 (Ω, and assume that the conditions
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationPDE & FEM TERMINOLOGY. BASIC PRINCIPLES OF FEM.
PDE & FEM TERMINOLOGY. BASIC PRINCIPLES OF FEM. Sergey Korotov Basque Center for Applied Mathematics / IKERBASQUE http://www.bcamath.org & http://www.ikerbasque.net 1 Introduction The analytical solution
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Variational Problems of the Dirichlet BVP of the Poisson Equation 1 For the homogeneous
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationChapter 1: The Finite Element Method
Chapter 1: The Finite Element Method Michael Hanke Read: Strang, p 428 436 A Model Problem Mathematical Models, Analysis and Simulation, Part Applications: u = fx), < x < 1 u) = u1) = D) axial deformation
More informationMath Tune-Up Louisiana State University August, Lectures on Partial Differential Equations and Hilbert Space
Math Tune-Up Louisiana State University August, 2008 Lectures on Partial Differential Equations and Hilbert Space 1. A linear partial differential equation of physics We begin by considering the simplest
More informationA very short introduction to the Finite Element Method
A very short introduction to the Finite Element Method Till Mathis Wagner Technical University of Munich JASS 2004, St Petersburg May 4, 2004 1 Introduction This is a short introduction to the finite element
More informationFinite Element Methods for Fourth Order Variational Inequalities
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2013 Finite Element Methods for Fourth Order Variational Inequalities Yi Zhang Louisiana State University and Agricultural
More informationAdaptive Finite Element Methods Lecture Notes Winter Term 2017/18. R. Verfürth. Fakultät für Mathematik, Ruhr-Universität Bochum
Adaptive Finite Element Methods Lecture Notes Winter Term 2017/18 R. Verfürth Fakultät für Mathematik, Ruhr-Universität Bochum Contents Chapter I. Introduction 7 I.1. Motivation 7 I.2. Sobolev and finite
More informationSolution of Non-Homogeneous Dirichlet Problems with FEM
Master Thesis Solution of Non-Homogeneous Dirichlet Problems with FEM Francesco Züger Institut für Mathematik Written under the supervision of Prof. Dr. Stefan Sauter and Dr. Alexander Veit August 27,
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationProjected Surface Finite Elements for Elliptic Equations
Available at http://pvamu.edu/aam Appl. Appl. Math. IN: 1932-9466 Vol. 8, Issue 1 (June 2013), pp. 16 33 Applications and Applied Mathematics: An International Journal (AAM) Projected urface Finite Elements
More informationLecture Notes: African Institute of Mathematics Senegal, January Topic Title: A short introduction to numerical methods for elliptic PDEs
Lecture Notes: African Institute of Mathematics Senegal, January 26 opic itle: A short introduction to numerical methods for elliptic PDEs Authors and Lecturers: Gerard Awanou (University of Illinois-Chicago)
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Nonconformity and the Consistency Error First Strang Lemma Abstract Error Estimate
More informationChapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems
Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Sobolev Embedding Theorems Embedding Operators and the Sobolev Embedding Theorem
More information1. Let a(x) > 0, and assume that u and u h are the solutions of the Dirichlet problem:
Mathematics Chalmers & GU TMA37/MMG800: Partial Differential Equations, 011 08 4; kl 8.30-13.30. Telephone: Ida Säfström: 0703-088304 Calculators, formula notes and other subject related material are not
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More information1 Discretizing BVP with Finite Element Methods.
1 Discretizing BVP with Finite Element Methods In this section, we will discuss a process for solving boundary value problems numerically, the Finite Element Method (FEM) We note that such method is a
More informationTD 1: Hilbert Spaces and Applications
Université Paris-Dauphine Functional Analysis and PDEs Master MMD-MA 2017/2018 Generalities TD 1: Hilbert Spaces and Applications Exercise 1 (Generalized Parallelogram law). Let (H,, ) be a Hilbert space.
More informationFinite Elements. Colin Cotter. January 15, Colin Cotter FEM
Finite Elements January 15, 2018 Why Can solve PDEs on complicated domains. Have flexibility to increase order of accuracy and match the numerics to the physics. has an elegant mathematical formulation
More informationFinite Element Method for Ordinary Differential Equations
52 Chapter 4 Finite Element Method for Ordinary Differential Equations In this chapter we consider some simple examples of the finite element method for the approximate solution of ordinary differential
More informationFinite Elements. Colin Cotter. January 18, Colin Cotter FEM
Finite Elements January 18, 2019 The finite element Given a triangulation T of a domain Ω, finite element spaces are defined according to 1. the form the functions take (usually polynomial) when restricted
More informationBasic Concepts of Adaptive Finite Element Methods for Elliptic Boundary Value Problems
Basic Concepts of Adaptive Finite lement Methods for lliptic Boundary Value Problems Ronald H.W. Hoppe 1,2 1 Department of Mathematics, University of Houston 2 Institute of Mathematics, University of Augsburg
More informationLinear Analysis Lecture 5
Linear Analysis Lecture 5 Inner Products and V Let dim V < with inner product,. Choose a basis B and let v, w V have coordinates in F n given by x 1. x n and y 1. y n, respectively. Let A F n n be the
More informationNUMERICAL PARTIAL DIFFERENTIAL EQUATIONS
NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS Lecture Notes, Winter 2011/12 Christian Clason January 31, 2012 Institute for Mathematics and Scientific Computing Karl-Franzens-Universität Graz CONTENTS I BACKGROUND
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationNumerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods
Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods Contents Ralf Hartmann Institute of Aerodynamics and Flow Technology DLR (German Aerospace Center) Lilienthalplatz 7, 3808
More informationMath 660-Lecture 15: Finite element spaces (I)
Math 660-Lecture 15: Finite element spaces (I) (Chapter 3, 4.2, 4.3) Before we introduce the concrete spaces, let s first of all introduce the following important lemma. Theorem 1. Let V h consists of
More informationNumerical Methods for the Navier-Stokes equations
Arnold Reusken Numerical Methods for the Navier-Stokes equations January 6, 212 Chair for Numerical Mathematics RWTH Aachen Contents 1 The Navier-Stokes equations.............................................
More informationChapter 12. Partial di erential equations Di erential operators in R n. The gradient and Jacobian. Divergence and rotation
Chapter 12 Partial di erential equations 12.1 Di erential operators in R n The gradient and Jacobian We recall the definition of the gradient of a scalar function f : R n! R, as @f grad f = rf =,..., @f
More informationDIFFERENTIAL EQUATIONS
NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS Lecture Notes, Winter 2013/14 Christian Clason July 15, 2014 Institute for Mathematics and Scientific Computing Karl-Franzens-Universität Graz CONTENTS I BACKGROUND
More informationThe Mortar Boundary Element Method
The Mortar Boundary Element Method A Thesis submitted for the degree of Doctor of Philosophy by Martin Healey School of Information Systems, Computing and Mathematics Brunel University March 2010 Abstract
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More information2. Dual space is essential for the concept of gradient which, in turn, leads to the variational analysis of Lagrange multipliers.
Chapter 3 Duality in Banach Space Modern optimization theory largely centers around the interplay of a normed vector space and its corresponding dual. The notion of duality is important for the following
More informationFinite Element Interpolation
Finite Element Interpolation This chapter introduces the concept of finite elements along with the corresponding interpolation techniques. As an introductory example, we study how to interpolate functions
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationApplied/Numerical Analysis Qualifying Exam
Applied/Numerical Analysis Qualifying Exam August 9, 212 Cover Sheet Applied Analysis Part Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless,
More informationOne-dimensional and nonlinear problems
Solving PDE s with FEniCS One-dimensional and nonlinear problems L. Ridgway Scott The Institute for Biophysical Dynamics, The Computation Institute, and the Departments of Computer Science and Mathematics,
More informationWELL POSEDNESS OF PROBLEMS I
Finite Element Method 85 WELL POSEDNESS OF PROBLEMS I Consider the following generic problem Lu = f, where L : X Y, u X, f Y and X, Y are two Banach spaces We say that the above problem is well-posed (according
More informationCONVERGENCE THEORY. G. ALLAIRE CMAP, Ecole Polytechnique. 1. Maximum principle. 2. Oscillating test function. 3. Two-scale convergence
1 CONVERGENCE THEOR G. ALLAIRE CMAP, Ecole Polytechnique 1. Maximum principle 2. Oscillating test function 3. Two-scale convergence 4. Application to homogenization 5. General theory H-convergence) 6.
More informationPART IV Spectral Methods
PART IV Spectral Methods Additional References: R. Peyret, Spectral methods for incompressible viscous flow, Springer (2002), B. Mercier, An introduction to the numerical analysis of spectral methods,
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationPREPRINT 2010:25. Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method ERIK BURMAN PETER HANSBO
PREPRINT 2010:25 Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method ERIK BURMAN PETER HANSBO Department of Mathematical Sciences Division of Mathematics CHALMERS
More informationFINITE ELEMENT METHODS
FINITE ELEMENT METHODS Lecture notes arxiv:1709.08618v1 [math.na] 25 Sep 2017 Christian Clason September 25, 2017 christian.clason@uni-due.de https://udue.de/clason CONTENTS I BACKGROUND 1 overview of
More informationFact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen.
More informationConvex Geometry. Carsten Schütt
Convex Geometry Carsten Schütt November 25, 2006 2 Contents 0.1 Convex sets... 4 0.2 Separation.... 9 0.3 Extreme points..... 15 0.4 Blaschke selection principle... 18 0.5 Polytopes and polyhedra.... 23
More informationPartial Differential Equations and the Finite Element Method
Partial Differential Equations and the Finite Element Method Pavel Solin The University of Texas at El Paso Academy of Sciences ofthe Czech Republic iwiley- INTERSCIENCE A JOHN WILEY & SONS, INC, PUBLICATION
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationA posteriori error estimation for elliptic problems
A posteriori error estimation for elliptic problems Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationFrom Completing the Squares and Orthogonal Projection to Finite Element Methods
From Completing the Squares and Orthogonal Projection to Finite Element Methods Mo MU Background In scientific computing, it is important to start with an appropriate model in order to design effective
More informationA metric space X is a non-empty set endowed with a metric ρ (x, y):
Chapter 1 Preliminaries References: Troianiello, G.M., 1987, Elliptic differential equations and obstacle problems, Plenum Press, New York. Friedman, A., 1982, Variational principles and free-boundary
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationBoundary Value Problems and Iterative Methods for Linear Systems
Boundary Value Problems and Iterative Methods for Linear Systems 1. Equilibrium Problems 1.1. Abstract setting We want to find a displacement u V. Here V is a complete vector space with a norm v V. In
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationIntroduction to finite element exterior calculus
Introduction to finite element exterior calculus Ragnar Winther CMA, University of Oslo Norway Why finite element exterior calculus? Recall the de Rham complex on the form: R H 1 (Ω) grad H(curl, Ω) curl
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationWeak Formulation of Elliptic BVP s
Weak Formulation of Elliptic BVP s There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed
More informationHamburger Beiträge zur Angewandten Mathematik
Hamburger Beiträge zur Angewandten Mathematik Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations Klaus Deckelnick and Michael
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationMaximum norm estimates for energy-corrected finite element method
Maximum norm estimates for energy-corrected finite element method Piotr Swierczynski 1 and Barbara Wohlmuth 1 Technical University of Munich, Institute for Numerical Mathematics, piotr.swierczynski@ma.tum.de,
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationNumerical Methods for Partial Differential Equations
Numerical Methods for Partial Differential Equations Eric de Sturler University of Illinois at Urbana-Champaign Read section 8. to see where equations of type (au x ) x = f show up and their (exact) solution
More informationA WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS
A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS LIN MU, JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. This article introduces and analyzes a weak Galerkin mixed finite element method
More informationAppendix A Functional Analysis
Appendix A Functional Analysis A.1 Metric Spaces, Banach Spaces, and Hilbert Spaces Definition A.1. Metric space. Let X be a set. A map d : X X R is called metric on X if for all x,y,z X it is i) d(x,y)
More informationApproximation in Banach Spaces by Galerkin Methods
2 Approximation in Banach Spaces by Galerkin Methods In this chapter, we consider an abstract linear problem which serves as a generic model for engineering applications. Our first goal is to specify the
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationMULTIGRID METHODS FOR MAXWELL S EQUATIONS
MULTIGRID METHODS FOR MAXWELL S EQUATIONS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationDomain Perturbation for Linear and Semi-Linear Boundary Value Problems
CHAPTER 1 Domain Perturbation for Linear and Semi-Linear Boundary Value Problems Daniel Daners School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia E-mail: D.Daners@maths.usyd.edu.au
More informationBasic Principles of Weak Galerkin Finite Element Methods for PDEs
Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element
More informationIterative Methods for Linear Systems
Iterative Methods for Linear Systems 1. Introduction: Direct solvers versus iterative solvers In many applications we have to solve a linear system Ax = b with A R n n and b R n given. If n is large the
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationDISCRETE MAXIMUM PRINCIPLES in THE FINITE ELEMENT SIMULATIONS
DISCRETE MAXIMUM PRINCIPLES in THE FINITE ELEMENT SIMULATIONS Sergey Korotov BCAM Basque Center for Applied Mathematics http://www.bcamath.org 1 The presentation is based on my collaboration with several
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationNumerische Mathematik
Numer. Math. (2012) 122:61 99 DOI 10.1007/s00211-012-0456-x Numerische Mathematik C 0 elements for generalized indefinite Maxwell equations Huoyuan Duan Ping Lin Roger C. E. Tan Received: 31 July 2010
More informationIsogeometric Analysis:
Isogeometric Analysis: some approximation estimates for NURBS L. Beirao da Veiga, A. Buffa, Judith Rivas, G. Sangalli Euskadi-Kyushu 2011 Workshop on Applied Mathematics BCAM, March t0th, 2011 Outline
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More information1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e.,
Abstract Hilbert Space Results We have learned a little about the Hilbert spaces L U and and we have at least defined H 1 U and the scale of Hilbert spaces H p U. Now we are going to develop additional
More informationYongdeok Kim and Seki Kim
J. Korean Math. Soc. 39 (00), No. 3, pp. 363 376 STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM Yongdeok Kim and Seki Kim Abstract. Stability result is obtained for
More information1 Continuity Classes C m (Ω)
0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +
More informationChapter 1. Introduction
Chapter 1 Introduction Functional analysis can be seen as a natural extension of the real analysis to more general spaces. As an example we can think at the Heine - Borel theorem (closed and bounded is
More informationarxiv: v3 [math.na] 8 Sep 2015
A Recovery-Based A Posteriori Error Estimator for H(curl) Interface Problems arxiv:504.00898v3 [math.na] 8 Sep 205 Zhiqiang Cai Shuhao Cao Abstract This paper introduces a new recovery-based a posteriori
More informationPreconditioned space-time boundary element methods for the heat equation
W I S S E N T E C H N I K L E I D E N S C H A F T Preconditioned space-time boundary element methods for the heat equation S. Dohr and O. Steinbach Institut für Numerische Mathematik Space-Time Methods
More informationBasic Properties of Metric and Normed Spaces
Basic Properties of Metric and Normed Spaces Computational and Metric Geometry Instructor: Yury Makarychev The second part of this course is about metric geometry. We will study metric spaces, low distortion
More informationOptimization and Optimal Control in Banach Spaces
Optimization and Optimal Control in Banach Spaces Bernhard Schmitzer October 19, 2017 1 Convex non-smooth optimization with proximal operators Remark 1.1 (Motivation). Convex optimization: easier to solve,
More informationFEniCS Course. Lecture 0: Introduction to FEM. Contributors Anders Logg, Kent-Andre Mardal
FEniCS Course Lecture 0: Introduction to FEM Contributors Anders Logg, Kent-Andre Mardal 1 / 46 What is FEM? The finite element method is a framework and a recipe for discretization of mathematical problems
More information