Interpolation in h-version finite element spaces

Interpolation in h-version finite element spaces Thomas Apel Institut für Mathematik und Bauinformatik Fakultät für Bauingenieur- und Vermessungswesen Universität der Bundeswehr München Chemnitzer Seminar zur Optimierung mit partiellen Differentialgleichungen Thomas Apel Interpolation in h-version finite element spaces 1 / 25

Plan of the talk 1 Interpolation operators 2 Local error estimates Local error estimates for isotropic elements Local error estimates for anisotropic elements Thomas Apel Interpolation in h-version finite element spaces 2 / 25

Nodal interpolation I Finite element in the sense of Ciarlet: (K, P K, N K ) K R d with non-empty interior and piecewise smooth boundary, P K is an n-dimensional linear space of functions defined on K ( shape functions ), N K = {N i,k } n i=1 is a basis of the dual space P K ( nodal variables ). Nodal basis {φ j,k } n j=1 of P K : N i,k (φ j,k ) = δ i,j, i, j = 1,..., n, Nodal interpolation operator: I K u := n N i,k (u)φ i,k. i=1 Thomas Apel Interpolation in h-version finite element spaces 4 / 25

Nodal interpolation II Nodal basis: N i,k (φ j,k ) = δ i,j, i, j = 1,..., n, (1) Nodal interpolation operator: I K u := n N i,k (u)φ i,k. i=1 Condition (1) yields I K φ j,k = n N i,k (φ j,k )φ i,k = i=1 n δ i,j φ i,k = φ j,k, j = 1,..., n, i=1 and thus I K φ = φ φ P K. (2) Thomas Apel Interpolation in h-version finite element spaces 5 / 25

Assumption: existence of a reference element There is a reference element ( ˆK, P ˆK, N ˆK ) with For each K T there is a bijective mapping with K = F K ( ˆK ), u P K iff û := u F K P ˆK, and F K : ˆx R d x = F K (ˆx) R d N i,k (u) = N i, ˆK (u F K ), i = 1,..., n, for all u for which the functionals are well defined. Advantage: (I K u) F K = I ˆK û allows to estimate the error on the reference element ˆK and to transform the estimates to K. Thomas Apel Interpolation in h-version finite element spaces 6 / 25

Typical mappings If possible, an affine mapping is chosen, F K (ˆx) = Aˆx + a with A R d d, a R d, otherwise the isoparametric mapping is used, F K (ˆx) = m a i ψ i, (ˆx), ˆK i=1 where the shape of K is determined by the positions of nodes a i R d and shape functions ψ i, ˆK with ψ i, ˆK (a j ) = δ i,j. Thomas Apel Interpolation in h-version finite element spaces 7 / 25

Global nodal interpolation operator I Finite element space via finite element mesh T : FE T := {v L 2 (Ω) : v K := v K P K K T and v K1, v K2 share the same nodal values on K 1 K 2 }, (3) Global interpolation operator I T : (I T u) K = I K (u K ) K T. Thomas Apel Interpolation in h-version finite element spaces 8 / 25

Global nodal interpolation operator II The dimension of FE T is denoted by N +. The set N T,+ = {N i } N+ i=1 is the union of all N K, K T, where common nodal variables of adjacent elements are counted only once. The global basis {φ j } N+ j=1 FE T satisfies N i (φ j ) = δ i,j, i, j = 1,..., N +. Global interpolation operator I T : N + I T u = N i (u)φ i. (4) i=1 Thomas Apel Interpolation in h-version finite element spaces 9 / 25

Quasi-interpolation I T acts only on sufficiently regular functions such that all functionals N i are well defined. The remedy is the definition of a quasi-interpolation operator Q T u = N N i (Π i u)φ i, (5) i=1 that means, we replace the function u in (4) by regularized functions Π i u. The index i indicates that we may use for each functional N i a different, locally defined averaging operator Π i. Thomas Apel Interpolation in h-version finite element spaces 10 / 25

Isotropic and anisotropic elements h K... diameter of K, ϱ K... diameter of the largest ball inscribed in K, γ K = h K /ϱ K... aspect ratio of K. Isotropic elements have moderate aspect ratio: easy to achieve in mesh generation, numerical analysis at moderate technical expense. Anisotropic elements have large aspect ratio. approximation of anisotropic features in functions, constants must be uniformly bounded in the aspect ratio. Thomas Apel Interpolation in h-version finite element spaces 12 / 25

Result Let ( ˆK, P ˆK, N ˆK ) be a reference element with P l 1 P ˆK, (6) N ˆK (C s ( ˆK )). (7) Assume that (K, P K, N K ) is affine equivalent to ( ˆK, P ˆK, N ˆK ). Let u W l,p (K ) with l N, p [1, ], such that W l,p ( ˆK ) C s ( ˆK ), i.e. l > s + d p, (8) and let m {0,..., l 1} and q [1, ] be such that Then the estimate holds. W l,p ( ˆK ) W m,q ( ˆK ). (9) u I K u W m,q (K ) C K 1/q 1/p h l K ϱ m K u W l,p (K ) (10) Thomas Apel Interpolation in h-version finite element spaces 14 / 25

Transformation to reference element Let F K (ˆx) = Aˆx + a be an affine mapping with K = F K ( ˆK ). If û W m,q ( ˆK ) then u = û F 1 K W m,q (K ) and u W m,q (K ) C K 1/q ϱ m K û W m,q ( ˆK ). (11) If u W l,p (K ) then û = u F K W l,p ( ˆK ) and û W l,p ( ˆK ) C K 1/p h l K u W l,p (K ). (12) The constants depend on the shape and size of ˆK. Thomas Apel Interpolation in h-version finite element spaces 15 / 25

Error estimate on the reference element Let ( ˆK, P ˆK, N ˆK ) be a reference element with P l 1 P ˆK, N ˆK (C s ( ˆK )). Let û W l,p ( ˆK ) with l N, p [1, ], such that W l,p ( ˆK ) C s ( ˆK ), i.e. l > s + d p, and let m {0,..., l 1} and q [1, ] be such that Then the estimate W l,p ( ˆK ) W m,q ( ˆK ). holds. û I ˆK û W m,q ( ˆK ) C û W l,p ( ˆK ) Thomas Apel Interpolation in h-version finite element spaces 16 / 25

Boundedness of the interpolation operator From (7) and (8) we obtain N i, ˆK (ˆv) C ˆv C s ( ˆK ) C ˆv W l,p ( ˆK ) and thus with φ i, ˆK W m,q ( ˆK C the boundedness of the interpolation ) operator, n I ˆK ˆv W m,q ( ˆK = ) N i, (ˆv)φ ˆK i, ˆK i=1 W m,q ( ˆK ) n N i, (ˆv) φ ˆK i, ˆK W m,q ( ˆK ) i=1 C ˆv W l,p ( ˆK ) where the constant depends not only on ˆK, s, m, q, l, and p, but also on N ˆK. Thomas Apel Interpolation in h-version finite element spaces 17 / 25

The Deny-Lions lemma (1953/54) Let the domain G R d, diam G = 1, be star-shaped with respect to a ball B G, and let l 1 be an integer and p [1, ] real. For each u W l,p (G) there is a w P l 1 such that u w W l,p (G) C u W l,p (G), (13) where the constant C depends only on d, l, and γ := diam G/diam B = 1/diam B. Thomas Apel Interpolation in h-version finite element spaces 18 / 25

Final proof Combining these estimates, choosing ŵ P l 1 according to the Deny-Lions Lemma, and using ŵ = I ˆK ŵ due to (6), we get û I ˆK û W m,q ( ˆK ) = (û ŵ) I ˆK (û ŵ) W m,q ( ˆK ) C û ŵ W l,p ( ˆK ) C û W l,p ( ˆK ). (14) Thomas Apel Interpolation in h-version finite element spaces 19 / 25

Introduction Anisotropic elements are characterized by a large aspect ratio γ K := h K /ϱ K. The estimate can be rewritten as u I K u W m,q (K ) C K 1/q 1/p h l K ϱ m K u W l,p (K ) u I K u W m,q (K ) C K 1/q 1/p h l m K γ m K u W l,p (K ), that means that the quality of this estimate deteriorates if m 1 and γ K 1. We wish to have at least an estimate of the form u I K u W m,q (K ) C K 1/q 1/p h l m K u W l,p (K ). Thomas Apel Interpolation in h-version finite element spaces 21 / 25

Negative example Consider the triangle K with the nodes ( h, 0), (h, 0), and (0, εh), and interpolate the function u(x 1, x 2 ) = x1 2 in the vertices with polynomials of degree one. Then I K u = h 2 ε 1 hx 2 and u I K u W 1,2 (K ) u W 2,2 (K ) ( 1 = h 6 + 1 ) 1/2 4 ε 2 = c ε h with c ε for ε 0 and c ε Cγ K. Thomas Apel Interpolation in h-version finite element spaces 22 / 25

Positive example Consider now the triangle with the nodes (0, 0), (h, 0), and (0, εh), and interpolate again the function u(x 1, x 2 ) = x1 2 in P 1. We get I K u = hx 1 and u I K u W 1,2 (K ) = 1 h u W 2,2 (K ) 12 where the constant is independent of ε. The desired estimate is valid although the element is anisotropic for small ε. Wir brauchen die Maximalwinkelbedingung! Thomas Apel Interpolation in h-version finite element spaces 23 / 25

Consideration of the reference element The estimate û I ˆK û W 1,2 ( ˆK ) C û W 2,2 ( ˆK ) can be transformed via x 1 = hˆx 1, x 2 = εhˆx 2 to the element K from the last example. We obtain ( (u I h K u) 2 x 1 ( Ch 2 2 u x1 2 L 2 (K ) 2 L 2 (K ) + ε 2 (u I K u) x 2 + ε 2 2 2 u x 1 x 2 2 L 2 (K ) L 2 (K ) The bound for the x 2 -derivative depends on ε 1. ) 1/2 + ε 4 2 u x 2 2 2 L 2 (K ) ) 1/2 Thomas Apel Interpolation in h-version finite element spaces 24 / 25

Consequence We need to prove on the reference element the sharper estimate ( (û I ˆK û) 2 û ˆx C 2 L 2 ( ˆK ) ˆx 1 ˆx 2 2 L 2 ( ˆK ) + 2 û ˆx 2 2 2 L 2 ( ˆK ) ) 1/2. Simple proof: Set ˆv = (û I ˆK û) ˆx 2 Then the desired estimate reduces to ˆv L 2 ( ˆK ) C ˆv W 1,2 ( ˆK ), which is valid since 1 0 v(0, x 2) dx 2 = 0. Thomas Apel Interpolation in h-version finite element spaces 25 / 25