✬ ✩ M AXIMUM - NORM A POSTERIORI ESTIMATES ON ANISOTROPIC MESHES Natalia Kopteva University of Limerick, Ireland partially supported by DAAD (Study Visit Grant for Senior Academics) and Science Foundation Ireland ✫ ✪
P ROBLEM ADDRESSED 1 For singularly perturbed semilinear reaction-diffusion equations − ε 2 △ u + f ( x, u ) = 0 where x ∈ Ω ⊂ R 2 , subject to u = 0 on ∂ Ω f ( x, u ) − f ( x, v ) ≥ C f [ u − v ] whenever u ≥ v , ε 2 + C f � 1 we look for residual-type a posteriori error estimates � �� � � � � ≤ function max x � error mesh, comp.sol-n x ∈ ¯ Ω in the maximum norm on anisotropic meshes
W HY ANISOTROPIC MESHES ?? 2 • Interpolation error bounds ⇒ anisotropic meshes are superior for layer solutions (a) Standard mesh. (b) Fine mesh. (c) Shape-regular refinement. (d) Anisotropic ref-nt. 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 1 1 1 1 0 0 0 0 0 0 0 0.5 0 0.5 0.5 0.5 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 0 0 0 0 1 1 1 1 �
W HY ANISOTROPIC MESHES ?? 3 • anisotropic meshes are superior for layer solutions 0.6 3 1 0.4 2 0.8 0.2 1 0.6 0 0 0.4 −0.2 −1 0.2 0.5 −0.4 1 0 1 −0.6 0 0 0.5 0.5 −0.5 0.2 0 0.4 0.6 −0.5 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0.8 −1 0 −1 1 (i) fine in layer regions ; coarse outside (ii) maximum mesh aspect ratio ∼ (layer width) − 1 ≫ 1 ✭✭✭✭✭✭✭✭ ✭ ❈ ❈ BUT theoretical difficulties within the FEM framework...
O UTLINE 4 Part 0 Perceptions & expectations t.b. adjusted for anisotropic meshes Part 1 A posteriori estimates on anisotropic meshes — Problem addressed (more detail) — Existing literature — Mesh assumptions + preview of results Part 2 A bit of analysis : 3 technical issues addressed 1. Application of a Scaled Trace theorem when estimating the Jump Resid- ual (”long” edges cause problems...) 2. Shaper bounds for the Interior Residual (by identifying connected paths of anisotropic nodes...) 3. Quasi-interpolants (of Cl´ ement/Scott-Zhang type) are not readily avail- able for general anisotropic meshes [Apel, Chapt. III]... Numerics. Current+future work (3d; non-singularly perturbed case...) Part 3
Part 0 P ERCEPTIONS & EXPECTATIONS ... 5 One Perception: the computed-solution error in the maximum norm is closely related to the corresponding interpolation error... • Quasi-uniform meshes, linear elements � u − u h � L ∞ (Ω) ≤ ln( C + ε/h ) inf χ ∈ S h � u − χ � L ∞ (Ω) – Schatz, Wahlbin, On the quasi-optimality in L ∞ of the ˚ H 1 -projection into −△ u = f , finite element spaces , Math. Comp. 1982: – Schatz, Wahlbin, On the finite element method for singularly perturbed − ε 2 △ u + au = f , reaction-diffusion problems ... , Math. Comp., 1983:
Part 0 P ERCEPTIONS & EXPECTATIONS ... 5 One Perception: the computed-solution error in the maximum norm is closely related to the corresponding interpolation error... • Quasi-uniform meshes, linear elements � u − u h � L ∞ (Ω) ≤ ln( C + ε/h ) inf χ ∈ S h � u − χ � L ∞ (Ω) – Schatz, Wahlbin, On the quasi-optimality in L ∞ of the ˚ H 1 -projection into −△ u = f , finite element spaces , Math. Comp. 1982: – Schatz, Wahlbin, On the finite element method for singularly perturbed − ε 2 △ u + au = f , reaction-diffusion problems ... , Math. Comp., 1983: • Strongly-anisotropic triangulations: no such result – BUT this is frequently considered a reasonable heuristic conjecture t.b. used in the anisotropic mesh adaptation (Hessian-related metrics...) – IN FACT, this is NOT true (see next)
P ERCEPTIONS & EXPECTATIONS ( CONTINUED ) 6 Example: − ε 2 △ u + u = 0 with u = e − x/ε exhibiting a sharp boundary layer Observation #1: Mass Lumping may be superior on anisotropic meshes Standard linear FEM Mass Lumping 1 10 -1 10 -1 ǫ =2 -8 ǫ =2 -8 ǫ =2 -16 ǫ =2 -16 10 -2 10 -2 ǫ =2 -24 ǫ =2 -24 ∼ N -2 ln 2 N ∼ N -2 ln 2 N 10 -3 10 -3 ∼ N -1 lnN ∼ N -1 lnN 10 -4 10 -4 10 -5 10 -5 0 0 1 10 -6 10 -6 10 1 10 2 10 3 10 1 10 2 10 3 1 10 -1 10 -1 ǫ =2 -8 ǫ =2 -16 10 -2 10 -2 ǫ =2 -24 ∼ N -2 ln 2 N 10 -3 10 -3 ∼ N -1 lnN ǫ =2 -8 10 -4 10 -4 ǫ =2 -16 ǫ =2 -24 10 -5 ∼ N -2 ln 2 N 10 -5 0 0 1 ∼ N -1 lnN 10 -6 10 -6 10 1 10 2 10 3 10 1 10 2 10 3 Here we use a Shishkin mesh: piecewise-uniform, DOF ≃ N 2 , mesh diameter ≃ N − 1 � u − u I � L ∞ (Ω) ≃ N − 2 ln 2 N ≃ DOF − 1 ln( DOF )
P ERCEPTIONS & EXPECTATIONS ( CONTINUED ) 7 Same Example: − ε 2 △ u + u = 0 with u = e − x/ε exhibiting a sharp boundary layer Observation #2: Convergence Rates may depend on the mesh structure (even for mass lumping), NOT ONLY on the interpolation error Standard linear FEM Mass Lumping 1 10 -1 10 -1 ǫ =2 -8 ǫ =2 -8 ǫ =2 -16 ǫ =2 -16 10 -2 10 -2 ǫ =2 -24 ǫ =2 -24 N -2 N -2 10 -3 10 -3 N -1 N -1 10 -4 10 -4 10 -5 10 -5 0 0 1 10 -6 10 -6 10 1 10 2 10 3 10 1 10 2 10 3 1 10 -1 10 -1 10 -2 10 -2 10 -3 10 -3 ǫ =2 -8 ǫ =2 -8 10 -4 ǫ =2 -16 10 -4 ǫ =2 -16 ǫ =2 -24 ǫ =2 -24 N -2 N -2 10 -5 10 -5 0 0 1 N -1 ǫ N -1 10 -6 10 -6 10 1 10 2 10 3 10 1 10 2 10 3 � u − u I � L ∞ (Ω) ≃ N − 2 ≃ DOF − 1 Here we use a graded Bakhvalov mesh:
W HAT GOES WRONG ?? 8 • A theoretical explanation of the above phenomena is given in: N.Kopteva, Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations , Math. Comp., 2014.
W HAT GOES WRONG ?? 8’ What happens in ˚ Ω := (0 , 2 ε ) × ( − H, H ) N 0 } 2 N 0 ω h := { x i = ε i i =0 × {− H, 0 , H } ?? with the tensor-product mesh ˚ H T 0 in Ω 0 ⊂ Ω : 0 T in Ω : −H ε ε 2 0 Mass lumping, U i := u h ( x i , 0) and U ± i := u h ( x i , ± H ) : h 2 [ − U i − 1 + 2 U i − U i +1 ] + ε 2 ε 2 H 2 [ − U − i + 2 U i − U + i ] + γ i U i = 0 γ N 0 = 2 with γ i = 1 for i � = N 0 , and 3 h 2 [ − U i − 1 + 2 U i − U i +1 ] + ε 2 ε 2 H 2 [ − U − i + 2 U i − U + ε ≪ H ⇒ i ] + γ i U i = 0
I MPLICATIONS 9 Implications of the above example: • Theoretical: if one tries to prove ”standard” (almost) second-order a priori/a posteriori er- ror estimate in the maximum norm on a general anisotropic mesh, this may be impossible... • Anisotropic mesh adaptation (Hessian-related metrics...): One needs to be careful with the heuristic conjecture that the computed-solution error in the maximum norm is closely related to the corresponding interpolation error...
P ERCEPTIONS & EXPECTATIONS ( CONTINUED ) 10 Non-singularly-perturbed EXAMPLE [Nochetto et al, Numer. Math., 2006]: −△ u + f ( u ) = 0 with f ( u ) ∼ − u − 3 and u = √ x 1 1 1 0 0 0 0 1 0 1 0 1 ǫ 10 -1 10 -1 10 -1 linearFE linearFE linearFE lumped-mass lumped-mass lumped-mass 10 -2 10 -2 10 -2 N -2 N -2 N -2 N -1.5 N -1.5 N -1.5 10 -3 10 -3 10 -3 10 -4 10 -4 10 -4 10 -5 10 -5 10 -5 10 -6 10 -6 10 -6 10 1 10 2 10 3 10 1 10 2 10 3 10 1 10 2 10 3 � u − u I � L ∞ (Ω) ≃ N − 2 ≃ DOF − 1 Graded mesh: { ( i/N ) 6 } N i =0 : Mesh transition parameter: ǫ = 0 . 1
O UTLINE 4 Part 0 Perceptions & expectations t.b. adjusted for anisotropic meshes Part 1 A posteriori estimates on anisotropic meshes — Problem addressed (more detail) — Existing literature — Mesh assumptions + preview of results Part 2 A bit of analysis : 3 technical issues addressed 1. Application of a Scaled Trace theorem when estimating the Jump Resid- ual (”long” edges cause problems...) 2. Shaper bounds for the Interior Residual (by identifying connected paths of anisotropic nodes...) 3. Quasi-interpolants (of Cl´ ement/Scott-Zhang type) are not readily avail- able for general anisotropic meshes [Apel, Chapt. III]... Numerics. Current+future work (3d; non-singularly perturbed case...) Part 3
P ART 1 P ROBLEM ADDRESSED ( DETAILS ) 11 For − ε 2 △ u + f ( x, u ) = 0 , we consider a standard finite element approximation ε 2 ( ∇ u h , ∇ v h ) + ( f I v h ∈ S h , f h := f ( · , u h ) , h , v h ) = 0 , where S h ⊂ H 1 0 (Ω) is a linear finite element space Ω is a polygonal, possibly non-Lipschitz, domain in R n , n = 2 : • 0 (Ω) ∩ C (¯ u ∈ H 1 ⇒ Ω) ; q ⊂ C (¯ Ω) for some l > 1 to be more precise, u ∈ W 2 l (Ω) ⊆ W 1 2 n and q > n . f u ( x, u ) ≥ C f ≥ 0 , • one-sided-Lipschitz-condition version of but f u ≤ ¯ C f NOT assumed
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