Overview A Posteriori rror stimates for the Biharmonic quation R Verfürth Fakultät für Mathematik Ruhr-Universität Bochum wwwruhr-uni-bochumde/num1 Milan / February 11th, 013 The Biharmonic quation Summary References 1/ 4 / 4 The Biharmonic quation The Biharmonic quation Variational Formulation and Discretization u = f u = 0 u n = 0 in on Γ on Γ Models the vertical displacement u of the mid-surface of a thin clamped plate under the influence of a vertical load f Models the vorticity of a two-dimensional Stokes flow Find u H0 () such that for all v H 0 () u v = fv Find u T X T H0 () such that for all v T X T u T v T = fv T Requires C 1 -elements 3/ 4 4/ 4
Proof of A Posteriori rror stimates Upper bound is standard Conformity X T H0 () implies Galerkin orthogonality Integration by parts twice element-wise yields L -representation Standard approximation properties of nodal interpolation prove upper bound Lower bound requires C 1 -cut-off functions z N λ z controls the element residual f T u T ψ,1 = ω z N λ,z χ ω controls the edge residual J (n( u T ) ψ,1 z N \N λ,z 1 ) z N 1 \N λ,z controls the edge residual J ( u T ) Smooth Cut-off Functions λ z λ,z λ,z (ψ ) ψ,1 5/ 4 6/ 4 A Posteriori rror stimates u u T h 4 + ft u T h J ( u T ) + h 3 J (n u T ) + h 4 f f T Mixed Variational Problem and Discretization Find ϕ H 1 () and u H0 1 () such that for all ψ H 1 () and v H 0 1 () ϕψ + ψ u = 0 v ϕ = fv Find ϕ T V T H 1 () and u T W T = V T H0 1 () such that for all ψ T V T and v T W T ϕ T ψ T + ψ T u T = 0 v T ϕ T = fv T The inf-sup condition is violated 7/ 4 8/ 4
Residuals Lower Bounds for R 1 R 1, ψ = = R, v = = ϕ T ψ + u T ψ (ϕ T u T ) ψ + ϕ T v + fv (f ϕ T ) v + J (n u T )ψ J (n ϕ T )v Test-functions ψ = ψ (ϕ T u T ), v = 0 yield ϕ T u T h 1 (u u T ) ϕ T u T + ϕ ϕ T ϕ T u T Test-functions ψ = ψ J (n u T ), v = 0 give h 1 J (n ϕ T ) (u u T ) ω + h ϕ ϕ T ω Mesh-dependent norm v + h ψ may be a candidate for measuring the error Causes problems with R which is linked to (ϕ ϕ T ) 9/ 4 10/ 4 Lower Bounds for R Test-functions ψ = 0, v = v = ψ,1 (f T u T ) and integration by parts for the (ϕ ϕ T )-term yield h 3 f T u T h ϕ ϕ T + h 3 f f T Test-functions ψ = 0, v = ψ,1 J (n ϕ T ) and integration by parts for the (ϕ ϕ T )-term give h 5 J (n ϕ T ) h ϕ ϕ T ω + h 3 f f T ω rror estimator h ϕ T u T + h J (n u T ) + h 6 f T ϕ T + h 5 J (n ϕ T ) 1 Upper Bound for (u u T ) Use a duality argument Choose g H 1 () such that g 1 = 1 and g, u u T = (u u T ) Denote by u g the solution of a biharmonic equation with right-hand side g; set ϕ g = u g Insert u u T as test-function in the mixed formulation of the auxiliary biharmonic equation Use the Galerkin orthogonality and standard approximation properties of the nodal interpolation to bound g, u u T in terms of the error estimator and ϕ g 1 + u g 3 Use the regularity result ϕ g 1 + u g 3 g 1 for convex domains 11/ 4 1/ 4
Upper Bound for } h ϕ ϕ T 1 A Posteriori rror stimates Introduce a mesh-function h W 1, () and assume that h h for every element Insert ψ = h (ϕ ϕ T ), v = h (u u T ) in the error equation to obtain h(ϕ ϕ T ) = R 1, h (ϕ ϕ T ) R, h (u u T ) h(ϕ ϕ T ) h (u u T ) + h(u u T ) (ϕ ϕ T ) h Bound the right-hand side using Galerkin orthogonality, interpolation error estimates, and inverse estimates h ϕ ϕ T + (u u T ) h ϕ T u T + h J (n u T ) + h 6 f T ϕ T + h 5 J (n ϕ T ) + h 6 f f T + max h f 1 13/ 4 14/ 4 Discretization Find u T S,0 0 (T ) (continuous, piecewise quadratic, vanishing on Γ) such that for all v T S,0 0 (T ) fv T = D u T : D v T + A (n D u T n )J (n v T ) + J (n u T )A (n D v T n ) + σ J (n u T )J (n v T ) Mesh-dependent Norm Mesh-dependent semi-norm v T T = v T ; Mesh-dependent norm v T T = u H0 () implies u u T T = v T T + σ u u T T +σ J (n v T ) J (n u T ) 15/ 4 16/ 4
Lifting to a Conforming Subspace of H 0() Hsieh-Clough-Tougher lement Compare u u T with u T u T where T u T is a lifting of u T to the Hsieh-Clough-Tougher subspace of H0 () T u T is defined by T u T (z) = u T (z), ( T u T )(z) = 1 u T (z), # ω z } ω z n ( T u T )(z ) = A (n u T )(z ) u u T T D (u T u T ) + T u T u T T v C 1 () : v i P 3 } function values, first order derivatives, normal derivatives 17/ 4 18/ 4 stimation of T u T u T T A local inverse estimate yields u T T u T T h 4 u T T u T A scaling argument gives h 4 u T T u T (u T T u T )(z) z N 1 + J (n u T ) The first term on the right-hand side can be controlled by the second one (same argument as for the ZZ-estimator) stimation of D (u T u T ) Since u T u T H0 () there is a v H 0 () with v = 1 and D (u T u T ) = D (u T u T ) : D v The term on the right-hand side can be rewritten and then estimated with the help of interpolation error estimates, inverse estimates, and trace inequalities 19/ 4 0/ 4
D (u T u T ) : D v = f(v i T v) D ( T u T u T ) : D v D u T : D (v i T v) + A (n D u T n )J (n i T v) + J (n u T )A (n D (i T v)n ) + σ J (n u T )J (n i T v) A Posteriori rror stimates u u T ; h 4 +σ + u T f T J (n u T ) h J ( u T ) + h 4 f f T 1/ 4 / 4 Summary References Summary References error sq estimator sq oscillation h 4 f T u T conf u u T +h J ( u T ) h 4 f f T A Charbonneau, Dossou, and R Pierre A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem Numer Meth PD 13 (1997), no 1, 93 111 +h 3 J (n u T ) h ϕ T u T mixed h(ϕ ϕ T ) +h J (n u T ) h 6 f f T + (u u T ) +h 6 f T ϕ T +h f 1 S C Brenner, T Gudi, and L-Y Sung An a posteriori error estimator for a quadratic C 0 -interior penalty method for the biharmonic problem IMA J Numer Anal 30 (010), no 3, 777 798 +h 5 J (n ϕ T ) h 4 f T u T nonconf u u T ;T +h J ( u T ) h 4 f f T +σ J (n u T ) 3/ 4 Handout of this talk wwwrubde/num1 A Posteriori rror stimation Techniques for Finite lement Methods Oxford University Press, 013 4/ 4