Traces and Duality Lemma
|
|
- Clinton Cook
- 5 years ago
- Views:
Transcription
1 Traces and Duality Lemma Recall the duality lemma with H / ( ) := γ 0 (H ()) defined as the trace space of H () endowed with minimal extension norm; i.e., for w H / ( ) L ( ), w H / ( ) = min{ ŵ H () ŵ H (), γ 0 ŵ = w}, H ( ) := dual to H / ( ) =: H / ( )! = γ ν (H(div, )). Any q H(div, ) (i.e. q L (, R ), div q L ()) defines γ ν q H ( ) by (γ ν q)(w) =: q ν, w = (q ŵ + ŵ div q)dx for w H / ( ) and ŵ H () with γ 0 ŵ = w. (Side note: implies γ ν q H ( ) q H(div,).) q ν, w q ŵ + div q ŵ q H(div,) ŵ H () Duality Lemma. (a) There exists exactly one such that for all q H (; R n ) γ ν L(H(div, ); H ( )) γ ν q = (γ 0 q) ν a.e. on. (b) Let, denote the duality brackets of H ( ) H / ( ). All q H(div, ) and v H () satisfy the formula ( ) γ ν q, γ 0 v = v div q + q v dx.
2 (c) The operator γ ν is surjective and ker γ ν = H 0 (div, ) := D(; R n ) H(div). (d) (Duality lemma) For all t H ( ) with t(w) =: t, w for w H / ( ), it holds t H ( ) = t, w w H / ( ), w H / = = v H (), γ 0 v 0 inf ϕ H () = inf q H(div,). q H(div,), γν q=t t, γ 0 v v ϕ H () Proof. Proof of (a). Let q H(div, ). denotes the unique weak solution of For all v H / ( ) ˆv H () ˆv + ˆv = 0 in, γ 0ˆv = v on. Then v H / ( ) = ˆv H (). Define X q (v) := (ˆv div q + q ˆv)dx The repeated application of the Cauchy Schwarz inequality shows X q (v) ˆv L () div q L () + q L () ˆv L () q H(div) ˆv H () = q H(div) v H ( ). Hence X q : H / ( ) R is linear and bounded. Thus for any q H(div, ) there exists g(q) H ( ) with X q = g(q). Define γ ν : q g(q). This operator is linear. The last inequality shows that the operator is also bounded, more precisely γ ν L(H(div,);H ( )). Moreover, for all functions v H / ( ) and q H (, R n ), an integration by parts leads to (γ 0 q) ν, v = (γ 0 q) ν, γ 0ˆv = (ˆv div q + q ˆv)dx,
3 Thus (γ 0 q) ν, v = γ ν q, v. Hence, for all v H / ( ) it holds (γ 0 q) ν γ ν q, v = 0, i.e., (γ 0 q) ν = γ ν q H ( ). This implies (a). Moreover,, extends the scalar product (, ) in L ( ) for smooth functions. Proof of (b). For all v H () and q H(div, ) it holds ( ) γ ν q, γ 0 v = q ˆv + ˆv div q dx, where ˆv H () is such that ˆv + ˆv = 0 and γ 0 v = γ 0ˆv in the weak sense. Since ker γ 0 = H0() and v ˆv H0(), this equals ( ) γ ν q γ 0 vds = q v + v div q dx and implies (b). Proof of (c). For all q D(; R n ), γ ν q = (γ 0 q) ν = 0 a.e. by (a). Hence, H 0 (div, ) = D(; R n ) H(div) ker γν. The proof of ker γ ν H 0 (div, ) is more technical and can be found in the literature, i.e., in [Girault, V. and Raviart, P. A., Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, Heidelberg, New York (986)]. It remains to show the surjectivity of γ ν. Given any t H ( ), the functional T : H () R, v t, γ 0 v is linear and bounded, written T H (). The Riesz representation z H () of T in the Hilbert space H () satisfies z, H () = T ( ). For ϕ D(), it follows for γ 0 ϕ = 0 that z, ϕ H () = T (ϕ) = t, γ 0 ϕ = 0. This proves z + z = 0 in the weak sense. In particular, q := z L (; R n ) and div z = z leads to div q = z L (). Hence, q H(div, ) and q H(div) = ( div q + q ) / = ( z + z ) / = z H () = T (H ()). 3
4 For any v H (), it follows ( ) γ ν q, γ 0 v = q v + v div q dx = = z, v H () = T (v) = t, γ 0 v. ( ) z v + vz dx This implies γ ν q t, γ 0 v = 0 for all v H (), which is γ ν q t = 0 in H ( ). Consequently, t = γ ν q R(γ ν ). Proof of (d). For any t H ( ) let z and q be as above in the proof of (c). Then t H ( ) = t, γ 0ˆv ˆv H () γ 0ˆv H / = ( ) with t, γ 0ˆv = γ ν z, γ 0ˆv = z, ˆv H () z H () ˆv H (). Since ˆv H () =, this implies t H ( ) z H (). Conversely, t, γ 0 z = z, z H () = z H () proves t H ( ) z H (). This concludes the proof and characterizes H ( ) completely. Primal PMP with test functions in H () without (BC) leads to b(u, t; v) = a(u, v) t, v! = F (v) for all v H (). (P) Theorem. u solves (PMP) (u, γ ν u) solves (P). Proof. v H0() implies t, v = 0. Hence u solves (PMP). Let u H0() solve (PMP), then p := u H(div, ) leads to t := γ ν p H ( ) so that, for all v H (), it follows it follows t, v = p ν, γ 0 (v) = ( p v + v div p)dx = a(u, v) F (v). }{{}}{{} = u = f Define H 0 (div, ) := {q H(div, ) γ ν q = 0}. 4
5 Interface trace spaces For a shape-regular triangulation T of R n into simplices define H ( T ) := {(t K ) K T K T H ( K) q H(div, ) K T, γ ν (q K ) = t K H ( K)} endowed with the norm (t K ) K T H ( T ) := min{ q H(div,) K T, γ ν (q K ) = t K } and H (T ) := {v L () K T, v K H (K)} = H (K) K T with (v K ) K T H (T ) := K T v K H (K). Given t (t K ) K T H ( T ) and v H (T ) define t, v T := K T t K, v K K. There exists q H(div, ) such that t K = γ ν (q K ) H ( K) for all K T and t, v T = K T (q v + v div q)dx = K (q NC v + v div q)dtx q H(div,) v H (T )! = t H ( T ) v H () Define { b: (H 0 () H ( T )) H (T ) R b(u, t; v) := ((u, t), v) a NC (u, v) t, v T 5
6 Theorem. u solves (PMP) and t = (t K ) K T = (γ ν ( u K )) K T if and only if (u, t) H0() H ( T ) solves for all v H (T ). b(u, t; v) = F (v) Proof. The Proof is left as an exercise. Remark. t, v T = 0 for v H 0(). inf- Condition This section is devoted to some immediate estimation for β > 0. Recall X := X X := H0() H ( T ), Y := H (T ) and the bounded bilinear form b : X Y R with b(u, t; v) = b((u, t), v) = a NC (u, v) t, v T (u, t) X, v Y. For any (u, t) S(X) and v S(Y ) the Cauchy-Schwarz inequality leads to b(u, t; v) u v NC + t H ( T ) v H (T ) u + t v H ( T ) NC + v H (T ). With v NC v H (T ) the choice of (u, t) and v finally shows, that b(u, t; v). Given (u, t) S(X) set M := b(u, t; ) H (T ). For u 0 choose v := u/ u H (T ) to obtain t, u T = ( ) t, u K = q u + u div q dx = u q νds = 0. K T Hence, b(u, t; v) = a NC(u, u) u H (T ) = u u + u. The Friedrichs inequality implies u C F () u with C F width()/π. This leads to b(u, t; v) u + C F () M. 6
7 Hence u M + CF (). () Given t let q H(div, ) have minimal extension norm in H(div, ) with q ν = t on K for all K T. The duality lemma leads to some v H (T ) with v H (T ) = and t H ( T ) = t, v T (i.e. v is the normed Riesz representation of t, T in H (T )). This implies whence u v NC + t H = a NC (u, v) + t H = b(u, t; v) M, The inequalities () and () show that = u + t H ( T ) M ( + This leads to t H ( T ) u M. () + C F ()) + M ( + C F ()). ( + C F ()) + + C F () M = b(u, t; ) H (T ). Since this holds for all (u, t) S(X), it implies 0 < 3 + C F () + + C F () β = inf x S(X) v S(H (T )) b(u, t; v). Splitting Lemmas Splitting Lemma I. Given real Hilbert spaces X, X, X := X X, {0} Y Y and bounded bilinear forms b j : X j Y for j =,, let b : X Y R, (x, x ; y) b (x, y) + b (x, y). Suppose (A) 0 < β := (A) 0 < β := (A3) b X Y = 0. inf x S(X ) y S(Y ) inf x S(X ) y S(Y ) b (x, y ), b (x, y), 7
8 Then b satisfies an inf- condition with β > 0 and 0 < β β (β + b ) + β β β. Example (Application to primal dpg for PMP). Let X := H 0(), X := H ( T ) and Y := H 0() H (T ) =: Y. Ad (A). Show that β = inf u H0 (), u = v H0 (), v + v = a(u, v) = + C F (). Proof of is as above. For utilize the first Dirichlet eigenpair (λ, Φ ) with Φ = and Φ = λ / so that C F () = λ and Φ H (T ) = + λ. Consequently β a(φ, v) v H0 (), v H () = Φ. Since the eigenvectors (Φ j ) j N form an L -orthonormal and a-orthogonal basis of H0() the remum is attained by v := Φ / Φ H (). This leads to β Φ Φ Φ H () = Φ Φ H () Ad (A). By duality lemma it holds β := inf t S(H ( T )) v S(H (T )) = t, v T = λ + λ = Ad (A3). The Cauchy-Schwarz inequality implies b = u S(H0 ()) v S(H (T )) a NC (u, v) = + λ + C F (). inf t H ( T ) =. t S(H ( T )) u S(H0 ()) v S(H (T )) u v. The proof of b is left as an exercise. This leads to the inf- estimate + C F () + ( + C F () + ) = 3 + C F () + + C F () β. 8
9 Proof of the first splitting lemma. Given (x, x ) S(X X ) let s := x X and x X = s for 0 s. Then (A) and (A3) imply β s b (x, ) Y = b(x, x ; ) Y =: M. Moreover, (A), the definition of b and triangle inequality show that β s b (x, ) Y b(x, x ; ) Y + b (x, ) Y M + b s. Consequently f(s) := max{β s, β s b s} M. It remains to compute min f := min 0 s f(s) M. Since (x, x ) S(X) is arbitrary, this lead to β 0 β. The monotony of β s and β s b s shows that the minimizer s exists in (0, ) with ( b + β )s = β s. Set κ := β /(β + b ), so s = κ ( s ), whence s = κ/ + κ. Consequently, concludes the proof. β 0 = β κ + κ Splitting Lemma II. In addition to the notation of the first splitting lemma with (A)-(A), pose Y := {y Y b (, y) = 0 in X } (then (A3) follows and characterizes maximal Y in (A3)) and Then and N := {y Y b (, y ) = 0 in X } = {0}. N := {y Y b(, y) = 0 in X} = 0 β := β β β + β + b + (β + β + b ) 4β β β. 9
10 Example (Application to primal dpg for PMP). Given v Y, then for any q H(div, ) follows 0 = (v div q + q NC v) dx. Hence, any α =, and ϕ H () satisfy 0 = (v ϕ/ α + ϕ e α NC v) dx. Hence, NC v is the weak gradient of v L (), i.e. v H (). Consequently, v q ν ds = 0 for all q H(div, ). This implies v = 0 on, whence v H 0(). Consequently, Moreover, Y = {v H (T ) t H ( T ), t, v T = 0} = H 0(). N := {w H 0() a NC (, w) = 0 in H 0()} = {0}. Recall = β = b and / + CF () and compute β 0 β = ( + CF ()) + + (3 + CF ()) 4( + CF ()) = 3 + C F () C F () + 8CF () Proof of the second splitting lemma. Since Y is a closed subspace of the Hilbert space Y, there is an orthogonal decomposition Y = Y Y with Y = Y. Then 0 < inf x S(X ) y S(Y ) b (x, y) = inf x S(X ) y S(Y ) b (x, y ) = β. Any y Y with b (, y ) = 0 in X belongs to Y, whence y Y Y = {0}. Consequently, b X Y satisfies inf- condition with β and is nondegenerate. General theory of bilinear forms shows β = inf y S(Y ) x S(X ) b (x, y ) > 0. 0
11 Given any (x, x ) S(X X ) there exists a unique solution y Y to b (, y ) = x, X. From Riesz isomorphism follows b(, y ) X = x X. Then for any y Y β y Y b (, y ) X = x X. Since β > 0 and N = {0}, b X Y satisfies inf- conditions and is nondegenerate, whence there exists a unique solution y Y to Consequently, Altogether b (, y ) =, x X b (, y ) in X. β Y Y b (, y ) X x X + b y Y. b(x, y + y ) = b (x, y + y ) + b (x, y + y ) On the other hand, = x X + b (x, y ) = x X + x Y =. y + y Y = y Y + y Y β ( x X + b x X /β ) + x X /β ( β = ( x X, x X ) b β ) ( ) β x X b β β β ( + b /β) x X is bounded from above by the maximal eigenvalue Λ of the matrix ( ) b /β β. b /β This implies β + b β Λ b(x, y + y ) y + y Y b(x, ) Y. Since x S(X) is arbitrary, this proves β Λ. The formula follows from explicit calculations of the above matrix.
12 Discretization Define for k N 0 S k+ 0 (T ) X = H 0() P k (E) X = H ( T ) X h := S k+ 0 P k (E) Y h := P k+d (T ) Y = H (T ). Suggest d = dimension of domain and all k N 0. This lecture studies d = for n = space dimensions and k = 0. Remark ( on P 0 (E) H ( T )). Given any t 0 P 0 (E). Let τ RT RT 0 (T ) H(div, ) satisfy E E : t 0 = τ RT ν E on E. Then for all v H (T ). t 0, v T = K T K (τ RT K ν K )v ds Discrete duality lemma. For any t 0 P 0 (E) there exists exactly one p RT RT 0 (T ) H(div, ) such that for all K T and E E(K) (ν E ν K E )t 0 = (p RT ν K ) E. Then t 0 H ( T ) p RT H(div,) + h max π t 0 H ( T ). Proof. Recall that t 0 H ( T ) is the minimum of all q H(div,) for any q H(div, ) with (ν E ν K E )t 0 = (q ν K ) E for all K T, E E(K). (3) This proves the first inequality. Given any q H(div, ), (p RT q) ν K = 0 on K defined by the integration-by-parts formula. In particular 0 = (p RT q) ν K ds = div(p RT q) dx for all K T. K K
13 Consequently, div p RT = Π 0 div q a.e. in. An integration by parts shows for any v H (T ) that (p RT q) NC v dx = (v Π 0 v) div(q p RT ) dx h max /π v NC ( Π 0 ) div q L (). Set v(x) := (Π 0 p RT ) (x mid(k)) + /4(div p RT ) x mid(k) with NC v = Π 0 p RT + /(div p RT ) + / div p RT ( mid(t )) = p RT in the previous estimate to deduce p RT L () = q p RT dx + (p RT q) NC v dx whence q L () p RT L () + h max π p RT L () ( Π 0 ) div q L (), (4) p RT L () q L () + h max π ( Π 0) div q L (). This and (4) imply with λ = h max /π p RT H(div,) ( q L () + λ ( Π 0 ) div q L ()) + Π 0 div q L () ( + λ ) q L () + ( + /λ )λ ( Π 0 ) div q L () ( + λ )( q L () + ( Π 0) div q L () + Π 0 div q ). In other words, p RT H(div,) / + h max/π is a lower bound of q H(div,) for all q with (3). By definition of t 0 H / ( T ) as the minimum, this shows p RT H(div,) + h max /π t 0 H ( T ). Annulation property for P := INC loc : H (T ) H (T ) projection onto P (T ) defined by I loc NCv K := E E(K) E (v K ) dsψ E K P (K) for any v H (T ), K T. 3
14 Given any v H (T ), ( P )v Y = v INC locv + v INC locv + κ h max v NC. Consequently, the Kato lemma implies P = P + κ h max. Mean value property of the gradients Π 0 NC INC locv for all v H (T ) leads to the annulation property K T K t 0 (v K I loc NCv K ) ds = t 0, v P v T. Hence, for all x h = (u c, t 0 ) X h and v H (T ), it follows b(x h, v P v) = a NC (u c, v P v) t 0, v P v T = 0. The abstract theory asserts discrete inf- condition with β P β h b. This shows β + κ h max β h. The a posteriori analysis involves F ( P ) Y computable but not of higher order. κ h T f L (T ), which is 4
Discontinuous Petrov-Galerkin Methods
Discontinuous Petrov-Galerkin Methods Friederike Hellwig 1st CENTRAL School on Analysis and Numerics for Partial Differential Equations, November 12, 2015 Motivation discontinuous Petrov-Galerkin (dpg)
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 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 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 informationGlowinski Pironneau method for the 3D ω-ψ equations
280 GUERMOND AND QUARTAPELLE Glowinski Pironneau method for the 3D ω-ψ equations Jean-Luc Guermond and Luigi Quartapelle 1 LIMSI CNRS, Orsay, France, and Dipartimento di Fisica, Politecnico di Milano,
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 informationINF-SUP CONDITION FOR OPERATOR EQUATIONS
INF-SUP CONDITION FOR OPERATOR EQUATIONS LONG CHEN We study the well-posedness of the operator equation (1) T u = f. where T is a linear and bounded operator between two linear vector spaces. We give equivalent
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
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 informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationSolutions of Selected Problems
1 Solutions of Selected Problems October 16, 2015 Chapter I 1.9 Consider the potential equation in the disk := {(x, y) R 2 ; x 2 +y 2 < 1}, with the boundary condition u(x) = g(x) r for x on the derivative
More informationAxioms of Adaptivity (AoA) in Lecture 3 (sufficient for optimal convergence rates)
Axioms of Adaptivity (AoA) in Lecture 3 (sufficient for optimal convergence rates) Carsten Carstensen Humboldt-Universität zu Berlin 2018 International Graduate Summer School on Frontiers of Applied and
More informationNumerische Mathematik
Numer. Math. (2003) 94: 195 202 Digital Object Identifier (DOI) 10.1007/s002110100308 Numerische Mathematik Some observations on Babuška and Brezzi theories Jinchao Xu, Ludmil Zikatanov Department of Mathematics,
More informationChapter 2 Finite Element Spaces for Linear Saddle Point Problems
Chapter 2 Finite Element Spaces for Linear Saddle Point Problems Remark 2.1. Motivation. This chapter deals with the first difficulty inherent to the incompressible Navier Stokes equations, see Remark
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 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 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 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 informationFunctional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32
Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More information1 Constrained Optimization
1 Constrained Optimization Let B be an M N matrix, a linear operator from the space IR N to IR M with adjoint B T. For (column vectors) x IR N and y IR M we have x B T y = Bx y. This vanishes for all y
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 information4 Sobolev spaces, trace theorem and normal derivative
4 Sobolev spaces, trace theorem and normal derivative Throughout, n will be a sufficiently smooth, bounded domain. We use the standard Sobolev spaces H 0 ( n ) := L 2 ( n ), H 0 () := L 2 (), H k ( n ),
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 informationGoal. Robust A Posteriori Error Estimates for Stabilized Finite Element Discretizations of Non-Stationary Convection-Diffusion Problems.
Robust A Posteriori Error Estimates for Stabilized Finite Element s of Non-Stationary Convection-Diffusion Problems L. Tobiska and R. Verfürth Universität Magdeburg Ruhr-Universität Bochum www.ruhr-uni-bochum.de/num
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 informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1
Manuscript submitted to AIMS Journals Volume 3, Number 2, May 29 Website: http://aimsciences.org pp. 1 XX ON THE EXISTENCE OF TRANSMISSION EIGENVALUES Andreas Kirsch1 University of Karlsruhe epartment
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More informationMULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday.
MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* DOUGLAS N ARNOLD, RICHARD S FALK, and RAGNAR WINTHER Dedicated to Professor Jim Douglas, Jr on the occasion of his seventieth birthday Abstract
More informationDiscontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
J A N U A R Y 0 1 8 P R E P R N T 4 7 4 Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints gor Voulis * and Arnold Reusken nstitut für Geometrie und Praktische
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 informationProjection Theorem 1
Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality
More informationMultigrid Methods for Saddle Point Problems
Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In
More informationAxioms of Adaptivity (AoA) in Lecture 2 (sufficient for optimal convergence rates)
Axioms of Adaptivity (AoA) in Lecture 2 (sufficient for optimal convergence rates) Carsten Carstensen Humboldt-Universität zu Berlin 2018 International Graduate Summer School on Frontiers of Applied and
More informationThe Dirichlet-to-Neumann operator
Lecture 8 The Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator plays an important role in the theory of inverse problems. In fact, from measurements of electrical currents at the surface
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationNORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
More informationFinite Element Methods for Maxwell Equations
CHAPTER 8 Finite Element Methods for Maxwell Equations The Maxwell equations comprise four first-order partial differential equations linking the fundamental electromagnetic quantities, the electric field
More informationWeierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN
Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN 2198-5855 On the divergence constraint in mixed finite element methods for incompressible
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationQUADRILATERAL H(DIV) FINITE ELEMENTS
QUADRILATERAL H(DIV) FINITE ELEMENTS DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK Abstract. We consider the approximation properties of quadrilateral finite element spaces of vector fields defined
More informationDiscontinuous Galerkin Methods
Discontinuous Galerkin Methods Joachim Schöberl May 20, 206 Discontinuous Galerkin (DG) methods approximate the solution with piecewise functions (polynomials), which are discontinuous across element interfaces.
More informationCHAPTER II HILBERT SPACES
CHAPTER II HILBERT SPACES 2.1 Geometry of Hilbert Spaces Definition 2.1.1. Let X be a complex linear space. An inner product on X is a function, : X X C which satisfies the following axioms : 1. y, x =
More informationDavid Hilbert was old and partly deaf in the nineteen thirties. Yet being a diligent
Chapter 5 ddddd dddddd dddddddd ddddddd dddddddd ddddddd Hilbert Space The Euclidean norm is special among all norms defined in R n for being induced by the Euclidean inner product (the dot product). A
More informationSupplementary Notes on Linear Algebra
Supplementary Notes on Linear Algebra Mariusz Wodzicki May 3, 2015 1 Vector spaces 1.1 Coordinatization of a vector space 1.1.1 Given a basis B = {b 1,..., b n } in a vector space V, any vector v V can
More informationA spacetime DPG method for acoustic waves
A spacetime DPG method for acoustic waves Rainer Schneckenleitner JKU Linz January 30, 2018 ( JKU Linz ) Spacetime DPG method January 30, 2018 1 / 41 Overview 1 The transient wave problem 2 The broken
More information1 Definition and Basic Properties of Compa Operator
1 Definition and Basic Properties of Compa Operator 1.1 Let X be a infinite dimensional Banach space. Show that if A C(X ), A does not have bounded inverse. Proof. Denote the unit ball of X by B and the
More informationThe Factorization Method for a Class of Inverse Elliptic Problems
1 The Factorization Method for a Class of Inverse Elliptic Problems Andreas Kirsch Mathematisches Institut II Universität Karlsruhe (TH), Germany email: kirsch@math.uni-karlsruhe.de Version of June 20,
More informationOn Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1
On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that
More informationADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS
J. OPERATOR THEORY 44(2000), 243 254 c Copyright by Theta, 2000 ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS DOUGLAS BRIDGES, FRED RICHMAN and PETER SCHUSTER Communicated by William B. Arveson Abstract.
More informationI teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition
More informationj=1 [We will show that the triangle inequality holds for each p-norm in Chapter 3 Section 6.] The 1-norm is A F = tr(a H A).
Math 344 Lecture #19 3.5 Normed Linear Spaces Definition 3.5.1. A seminorm on a vector space V over F is a map : V R that for all x, y V and for all α F satisfies (i) x 0 (positivity), (ii) αx = α x (scale
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 information1 Inner Product Space
Ch - Hilbert Space 1 4 Hilbert Space 1 Inner Product Space Let E be a complex vector space, a mapping (, ) : E E C is called an inner product on E if i) (x, x) 0 x E and (x, x) = 0 if and only if x = 0;
More informationMixed Finite Element Methods. Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016
Mixed Finite Element Methods Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016 Linear elasticity Given the load f : Ω R n, find the displacement u : Ω R n and the
More informationNumerische Mathematik
Numer. Math. (006) 105:159 191 DOI 10.1007/s0011-006-0031-4 Numerische Mathematik Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface
More informationA note on the σ-algebra of cylinder sets and all that
A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In
More informationEXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR
More informationDivergence-conforming multigrid methods for incompressible flow problems
Divergence-conforming multigrid methods for incompressible flow problems Guido Kanschat IWR, Universität Heidelberg Prague-Heidelberg-Workshop April 28th, 2015 G. Kanschat (IWR, Uni HD) Hdiv-DG Práha,
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
MATHEMATICS OF COMPUTATION Volume 75, Number 256, October 2006, Pages 1659 1674 S 0025-57180601872-2 Article electronically published on June 26, 2006 ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED
More informationHilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality
(October 29, 2016) Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are
More information4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan
The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,
More informationLectures on. Weak Solutions of Elliptic Boundary Value Problems
Lectures on Weak Solutions of Elliptic Boundary Value Problems S. Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600 113. e mail: kesh@imsc.res.in 2 1 Some abstract variational
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 Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationMATH Linear Algebra
MATH 4 - Linear Algebra One of the key driving forces for the development of linear algebra was the profound idea (going back to 7th century and the work of Pierre Fermat and Rene Descartes) that geometry
More informationA Finite Element Method for the Surface Stokes Problem
J A N U A R Y 2 0 1 8 P R E P R I N T 4 7 5 A Finite Element Method for the Surface Stokes Problem Maxim A. Olshanskii *, Annalisa Quaini, Arnold Reusken and Vladimir Yushutin Institut für Geometrie und
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)
More informationA variation on the infsup condition
Cinquième foire européenne aux éléments finis Centre International de Rencontres Mathématiques Marseille Luminy, 17-19 mai 2007 A variation on the infsup condition François Dubois CNAM Paris and University
More informationInequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz
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 informationFunctional Analysis (2006) Homework assignment 2
Functional Analysis (26) Homework assignment 2 All students should solve the following problems: 1. Define T : C[, 1] C[, 1] by (T x)(t) = t x(s) ds. Prove that this is a bounded linear operator, and compute
More informationCHAPTER VIII HILBERT SPACES
CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugate-linear transformation if it is a reallinear transformation from X into Y, and if T (λx)
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 informationA posteriori error estimates for Maxwell Equations
www.oeaw.ac.at A posteriori error estimates for Maxwell Equations J. Schöberl RICAM-Report 2005-10 www.ricam.oeaw.ac.at A POSTERIORI ERROR ESTIMATES FOR MAXWELL EQUATIONS JOACHIM SCHÖBERL Abstract. Maxwell
More informationNew Helmholtz-Weyl decomposition in L r and its applications to the mathematical fluid mechanics
New Helmholtz-Weyl decomposition in L r and its applications to the mathematical fluid mechanics Hideo Kozono Mathematical Institute Tohoku University Sendai 980-8578 Japan Taku Yanagisawa Department of
More informationMIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS
MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS DOUGLAS N. ARNOLD, RICHARD S. FALK, AND JAY GOPALAKRISHNAN Abstract. We consider the finite element solution
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationOn Riesz-Fischer sequences and lower frame bounds
On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More information1 The Stokes System. ρ + (ρv) = ρ g(x), and the conservation of momentum has the form. ρ v (λ 1 + µ 1 ) ( v) µ 1 v + p = ρ f(x) in Ω.
1 The Stokes System The motion of a (possibly compressible) homogeneous fluid is described by its density ρ(x, t), pressure p(x, t) and velocity v(x, t). Assume that the fluid is barotropic, i.e., the
More informationAxioms of Adaptivity
Axioms of Adaptivity Carsten Carstensen Humboldt-Universität zu Berlin Open-Access Reference: C-Feischl-Page-Praetorius: Axioms of Adaptivity. Computer & Mathematics with Applications 67 (2014) 1195 1253
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More informationProblem of Second grade fluids in convex polyhedrons
Problem of Second grade fluids in convex polyhedrons J. M. Bernard* Abstract This article studies the solutions of a three-dimensional grade-two fluid model with a tangential boundary condition, in a polyhedron.
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 informationRiemannian geometry of the twistor space of a symplectic manifold
Riemannian geometry of the twistor space of a symplectic manifold R. Albuquerque rpa@uevora.pt Departamento de Matemática, Universidade de Évora Évora, Portugal September 004 0.1 The metric In this short
More informationA NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION
A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION JOHNNY GUZMÁN, ABNER J. SALGADO, AND FRANCISCO-JAVIER SAYAS Abstract. The analysis of finite-element-like Galerkin discretization techniques for the
More informationUniform inf-sup condition for the Brinkman problem in highly heterogeneous media
Uniform inf-sup condition for the Brinkman problem in highly heterogeneous media Raytcho Lazarov & Aziz Takhirov Texas A&M May 3-4, 2016 R. Lazarov & A.T. (Texas A&M) Brinkman May 3-4, 2016 1 / 30 Outline
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 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 informationMIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS
MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS JUHO KÖNNÖ, DOMINIK SCHÖTZAU, AND ROLF STENBERG Abstract. We derive new a-priori and a-posteriori error estimates for mixed nite
More informationSPECTRAL REPRESENTATIONS, AND APPROXIMATIONS, OF DIVERGENCE-FREE VECTOR FIELDS. 1. Introduction
SPECTRAL REPRESENTATIONS, AND APPROXIMATIONS, OF DIVERGENCE-FREE VECTOR FIELDS GILES AUCHMUTY AND DOUGLAS R. SIMPKINS Abstract. Special solutions of the equation for a solenoidal vector field subject to
More informationKonstantinos Chrysafinos 1 and L. Steven Hou Introduction
Mathematical Modelling and Numerical Analysis Modélisation Mathématique et Analyse Numérique Will be set by the publisher ANALYSIS AND APPROXIMATIONS OF THE EVOLUTIONARY STOKES EQUATIONS WITH INHOMOGENEOUS
More informationFractional Operators with Inhomogeneous Boundary Conditions: Analysis, Control, and Discretization
Fractional Operators with Inhomogeneous Boundary Conditions: Analysis, Control, and Discretization Harbir Antil Johannes Pfefferer Sergejs Rogovs arxiv:1703.05256v2 [math.na] 11 Sep 2017 September 12,
More informationA Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions
A Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions Zhiqiang Cai Seokchan Kim Sangdong Kim Sooryun Kong Abstract In [7], we proposed a new finite element method
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
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 information