MATH 676 Finite element methods in scientifjc computing Wolfgang - PowerPoint PPT Presentation

MATH 676 – Finite element methods in scientifjc computing Wolfgang Bangerth, T exas A&M University http://www.dealii.org/ Wolfgang Bangerth

Lecture 17.75: Generating adaptively refjned meshes: “A posteriori” error estimators http://www.dealii.org/ Wolfgang Bangerth

Adaptive mesh refjnement (AMR) Adaptive mesh refjnement happens in a loop: SOLVE → ESTIMATE → MARK → REFINE ● SOLVE: Assemble linear system, solve it ● ESTIMATE: Compute a refjnement indicator for each cell ● MARK: Determine which cells should be refjned ● REFINE: Refjne these cells (and possibly others) http://www.dealii.org/ Wolfgang Bangerth

The ESTIMATE phase ESTIMATE phase: Compute a “refjnement indicator” for each cell K : η = {η K 1 , η K 2 , η K 3 , ... , η K N } Recall from lecture 17.25: A “reasonable” approach is to estimate the interpolation error of the solution, which leads to the Kelly indicator: 1 / 2 1 / 2 ( ∫ ∂ K |[∇ u h ]| 2 ) η K = h K http://www.dealii.org/ Wolfgang Bangerth

Ideas The “ideal” refjnement indicator would be the actual error e K : 1 / 2 η K : = e K = ‖ u − u h ‖ K = ( ∫ K | u − u h | 2 dx ) Questions/problems: 1.Would a difgerent norm be more appropriate? 2.But we don't have the exact solution u(x) ! 3.Does refjning a cell decrease the overall error? http://www.dealii.org/ Wolfgang Bangerth

Ideas Question 3: Does refjning a cell decrease the overall error? Answers: ● It depends on the norm in which we measure the error ● For interpolation , refjning one cell only afgects the error on this cell ● For the Galerkin projection , refjning one cell afgects the error everywhere This is called “error pollution”. http://www.dealii.org/ Wolfgang Bangerth

Ideas Question 3: Does refjning a cell decrease the overall error? Consider the Laplace equation: ● Let T be a mesh, T + be any refjnement ● Let V h be the FE space on T , V h + be the FE space on T + ● Then + V h ⊂ V h ● Recall E : = ‖∇ ( u − u h )‖ = min v h ∈ V h ‖∇ ( u − v h )‖ E + : = ‖∇ ( u − u h + )‖ = min v h ∈ V h + ‖∇( u − v h )‖ ● Consequently: + ≤ E E http://www.dealii.org/ Wolfgang Bangerth

Ideas Question 3: Does refjning a cell decrease the overall error? But: ● It is not necessarily true that if + ≤ E E : = ‖ u − u h ‖ E ● The improvement may be marginal T o show that one needs to refjne enough + ≤ θ E, θ< 1 E cells. Strategy: Dörfmer marking / bulk refjnement marks the cells with the largest errors so that ∑ K ∈ marked cells η K ≥ α ∑ K ∈ T η K , 0 <α≤ 1 http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation Question 2: For evaluating the true error 1 / 2 η K : = e K = ‖ u − u h ‖ K = ( ∫ K | u − u h | 2 dx ) we would need to know the exact solution u(x) . Answer: Yes. This is the central problem. We need to fjnd formulas that help us estimate the error only in term of u h (x). This is called “a posteriori” error estimation. http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Consider −Δ u = f , u ∣ ∂Ω = 0. Start with Galerkin orthogonality: (∇ u, ∇ v )=( f ,v ) ∀ v ∈ V (∇ u h , ∇ v h )=( f ,v h ) ∀ v h ∈ V h ⊂ V ⏟ (∇( u − u h ) , ∇ v h )= 0 ∀ v h ∈ V h ⊂ V = : e http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Then consider the energy norm of the error: 2 = (∇ e , ∇ e ) Ω = (∇ e, ∇( e − I h e )) Ω ‖∇ e ‖ Ω = ∑ K (∇ e, ∇( e − I h e )) K = ∑ K (−Δ e,e − I h e ) K +( n ⋅∇ e,e − I h e ) ∂ K Now use the following facts: ⏟ −Δ e = −Δ( u − u h ) = −Δ u +Δ u h = f +Δ u h cell residual ⏟ ⏟ ⏟ ( e − I h e )∣ ∂Ω = (( u − u h )−( I h u − I h u h ))∣ ∂ Ω = u ∣ ∂Ω − I h u ∣ ∂ Ω = 0 u h = 0 http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Thus: 2 = ∑ K (−Δ e,e − I h e ) K +( n ⋅∇ e ,e − I h e ) ∂ K ‖∇ e ‖ Ω = ∑ K ( f +Δ u h ,e − I h e ) K +( n ⋅∇ e,e − I h e ) ∂ K ∖∂Ω The second term only integrates over interior faces. We visit each face twice (from K and from its neighbor K' ): 1 ∑ K ( n K ⋅∇ e ∣ K ,e − I h e ) ∂ K ∖∂ Ω = ∑ K 2 ( n K ⋅∇ e ∣ K ,e − I h e ) ∂ K ∖∂ Ω + 1 ⏟ 2 ( n K ' ⋅∇ e ∣ K ' ,e − I h e ) ∂ K ∖∂ Ω =− n K 1 = ∑ K 2 ( n K ⋅ (∇ e ∣ K −∇ e ∣ K ' ) ,e − I h e ) ∂ K ∖∂Ω http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Consider the face term: 1 ∑ K ( n K ⋅∇ e ∣ K ,e − I h e ) ∂ K ∖∂ Ω = ∑ K 2 ( n K ⋅ (∇ e ∣ K −∇ e ∣ K ' ) ,e − I h e ) ∂ K ∖∂Ω Observe: ⏟ ⏟ ∇ e ∣ K −∇ e ∣ K ' = (∇ u ∣ K −∇ u ∣ K ' ) −(∇ u h ∣ K −∇ u h ∣ K ' ) = 0 because u is smooth = : [∇ u h ] http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Thus: ,e − I h e ) K − 1 2 = ∑ K ( f +Δ u h ⏟ ⏟ ‖∇ e ‖ 2 ( n ⋅ [∇ u h ] ,e − I h e ) ∂ K ∖∂Ω Ω cell residual "jump" residual Using the Cauchy-Schwarz inequality on cell/face integrals: 2 ≤ ∑ K ( ‖ f +Δ u h ‖ ∂ K ∖∂Ω ) K ‖ e − I h e ‖ K + 1 ‖∇ e ‖ 2 ‖ n ⋅[∇ u h ]‖ ∂ K ∖∂ Ω ‖ e − I h e ‖ Ω http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Thus: ,e − I h e ) K − 1 2 = ∑ K ( f +Δ u h ⏟ ⏟ ‖∇ e ‖ 2 ( n ⋅ [∇ u h ] ,e − I h e ) ∂ K ∖∂Ω Ω cell residual "jump" residual ≤ ∑ K ( ‖ f +Δ u h ‖ K ‖ e − I h e ‖ ∂ K ∖∂Ω ) K + 1 2 ‖ n ⋅ [∇ u h ]‖ ∂ K ∖∂Ω ‖ e − I h e ‖ 1 Because we have the following basic e ∈ H interpolation estimates: ‖ e − I h e ‖ K ≤ C h K ‖∇ e ‖ K 1 / 2 ‖∇ e ‖ K ‖ e − I h e ‖ ∂ K ∖∂Ω ≤ C h K http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Thus: 2 ≤ ∑ K ( 1 / 2 ‖∇ e ‖ K ) + 1 ⏟ ⏟ ‖∇ e ‖ ‖ f +Δ u h ‖ K ‖ e − I h e ‖ 2 ‖ n ⋅[∇ u h ]‖ ∂ K ∖∂Ω ‖ e − I h e ‖ Ω ∂ K ∖∂ Ω K ≤ Ch K ‖∇ e ‖ ≤ Ch K K ≤ C ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ∂ K ∖∂Ω ) ‖∇ e ‖ 1 / 2 ‖ n ⋅[∇ u h ]‖ 2 h K K 1 / 2 ( ∑ y i 1 / 2 ∑ i x i y i ≤ ( ∑ x i 2 ) 2 ) Recall Cauchy-Schwarz inequality: Apply this to the sum over cells: 2 ≤ C [ ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ] 2 ] 1 / 2 ∂ K ∖∂Ω ) 2 1 / 2 [ ∑ K ‖∇ e ‖ K 1 / 2 ‖ n ⋅[∇ u h ]‖ ‖∇ e ‖ 2 h K ⏟ Ω ‖∇ e ‖ Ω http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Thus: Ω ≤ C [ ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ] 1 / 2 ∂ K ∖∂Ω ) 2 1 / 2 ‖ n ⋅[∇ u h ]‖ ‖∇ e ‖ 2 h K Interpretation: We have bounded the error in terms of only the (computable!) discrete solution! http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation A posteriori error estimation for Laplace's equation: Last reformulation, squaring both sides: 2 ≤ C ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ∂ K ∖∂Ω ) 2 1 / 2 ‖ n ⋅ ‖∇ e ‖ 2 h K [∇ u h ]‖ Ω ⏟ 2 = : η K Nomenclature: ● We call the “residual-based” error estimator for η K cell K . ● It consists of the norm of the “cell residual” and the norm of the “jump residual”/”face residual”. http://www.dealii.org/ Wolfgang Bangerth

A posteriori error estimation Note: If we approximate 2 ≤ C ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ∂ K ∖∂Ω ) 2 1 / 2 ‖ n ⋅ ‖∇ e ‖ 2 h K [∇ u h ]‖ Ω ≈ C ' ∑ K h K ‖ n ⋅[∇ u h ]‖ 2 ∂ K ∖∂Ω then we get exactly the “Kelly” estimator! Reason: For odd polynomial degrees the jump residual dominates the cell residual. But: For even polynomial degrees, it is the other way around! http://www.dealii.org/ Wolfgang Bangerth

Some conclusions Conclusion 1: It is possible to bound the (unknown) error exclusively in terms of the known, computed solution. For example, for the Laplace equation we get: 2 ≤ C ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ∂ K ∖∂ Ω ) 2 1 / 2 ‖ n ⋅ ‖∇ e ‖ [∇ u h ]‖ 2 h K Ω The estimate involves “cell” and “jump” residuals. Note: Similar estimates can often be derived for other equation as well! http://www.dealii.org/ Wolfgang Bangerth

Some conclusions Conclusion 2: Estimates typically involve unknown constants. For example, for the Laplace equation we get: 2 ≤ C ∑ K ( h K ‖ f +Δ u h ‖ K + 1 ∂ K ∖∂ Ω ) 2 1 / 2 ‖ n ⋅ ‖∇ e ‖ [∇ u h ]‖ 2 h K Ω Here, the constant comes from the interpolation estimates. It is usually poorly known. Note: For other equations, there are also stability constants, and constants from yet other estimates. http://www.dealii.org/ Wolfgang Bangerth