A DPG method for the Schr odinger equation Jay Gopalakrishnan - PowerPoint PPT Presentation

A DPG method for the Schr¨ odinger equation Jay Gopalakrishnan Portland State University Collaborators: L. Demkowicz, S. Nagaraj, P. Sep´ ulveda RICAM Workshop on spacetime methods November 2016 AFOSR, NSF, RICAM Thanks: Jay Gopalakrishnan 1/28

“Petrov-Galerkin” schemes (PG) PG schemes are distinguished by different trial and test (Hilbert) spaces. � P.D.E.+ The problem: boundary conditions. ↓  Find x in a trial space X satisfying  Variational form: b ( x , y ) = ℓ ( y )  for all y in a test space Y. ↓  Find x h in a discrete trial space X h ⊂ X satisfying  Discretization: b ( x h , y h ) = ℓ ( y h )  for all y h in a discrete test space Y h ⊂ Y . For PG schemes, X h � = Y h in general. Jay Gopalakrishnan 2/28

� Elements of theory Variational formulation:   Exact inf-sup condition � � a uniqueness   | b ( x , y ) |  + = ⇒ wellposedness  C � x � X ≤ sup condition � y � Y y ∈ Y Babuˇ ska’s theorem:   Discrete inf-sup condition    = ⇒ � x − x h � X ≤ C inf � x − w h � X . | b ( x h , y h ) |  C � x h � X ≤ sup w h ∈ X h � y h � Y y h ∈ Y h Difficulty: Exact inf-sup condition = ⇒ Discrete inf-sup condition Jay Gopalakrishnan 3/28

� Elements of theory Variational formulation:   Exact inf-sup condition � � a uniqueness   | b ( x , y ) |  + = ⇒ wellposedness  C � x � X ≤ sup condition � y � Y y ∈ Y Babuˇ ska’s theorem:   Discrete inf-sup condition    = ⇒ � x − x h � X ≤ C inf � x − w h � X . | b ( x h , y h ) |  C � x h � X ≤ sup w h ∈ X h � y h � Y y h ∈ Y h Difficulty: Exact inf-sup condition = ⇒ Discrete inf-sup condition Is there a way to find a stable test space for any given trial space? Jay Gopalakrishnan 3/28

The ideal DPG method Pick any X h ⊆ X . The ideal DPG method finds x h ∈ X h such that ∀ y ∈ Y opt def b ( x h , y ) = ℓ ( y ) , = T ( X h ) , h where T : X �→ Y is defined by ( Tw , y ) Y = b ( w , y ) , ∀ w ∈ X , y ∈ Y . [ Demkowicz+G 2011 ] Rationale: Jay Gopalakrishnan 4/28

The ideal DPG method Pick any X h ⊆ X . The ideal DPG method finds x h ∈ X h such that ∀ y ∈ Y opt def b ( x h , y ) = ℓ ( y ) , = T ( X h ) , h where T : X �→ Y is defined by ( Tw , y ) Y = b ( w , y ) , ∀ w ∈ X , y ∈ Y . [ Demkowicz+G 2011 ] Rationale: Which function y maximizes | b ( x , y ) | Q: for any given x ? � y � Y A: y = Tx is the maximizer. ← Optimal test function. DPG Idea: If the discrete test space contains the optimal test functions, exact inf-sup condition = ⇒ discrete inf-sup condition . Jay Gopalakrishnan 4/28

The ideal DPG method Pick any X h ⊆ X . The ideal DPG method finds x h ∈ X h such that ∀ y ∈ Y opt def b ( x h , y ) = ℓ ( y ) , = T ( X h ) , h where T : X �→ Y is defined by ( Tw , y ) Y = b ( w , y ) , ∀ w ∈ X , y ∈ Y . It is important to (re)formulate the problem so that Y admits DG functions. Jay Gopalakrishnan 4/28

Quasi-Optimality of DPG Methods Assumption [U] (Uniqueness) { w ∈ X : b ( w , y ) = 0 ∀ y ∈ Y } = { 0 } . Assumption [I] (Inf-Sup & Continuity) | b ( w , y ) | ∃ C 1 , C 2 > 0 such that C 1 � y � Y ≤ sup ≤ C 2 � y � Y . � w � X w ∈ X Theorem [QO] (Quasi-Optimality) � x − x h � X ≤ C 2 Assumptions [U+I] = ⇒ inf � x − w h � X . C 1 w h ∈ X h [ Demkowicz+G 2011 ] Jay Gopalakrishnan 5/28

Next Quick introduction to DPG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Unbounded operator equations Broken weak form Heat equation vs. Schr¨ odinger equation A density result DPG method for the Schr¨ odinger equation Jay Gopalakrishnan 6/28

An unbounded operator setting We are interested in numerically solving operator equations Au = f where A : L 2 ( Ω ) m → L 2 ( Ω ) l is an (unbounded) operator satisfying [ Au ] i = ∂ α ( a ij α u j ) , a ij α : Ω → C are bounded functions ∀ i = 1 , . . . l , ∀ j = 1 , . . . , m , ∀ multi-indices | α | ≤ k , D ( Ω ) m ⊆ dom A . Jay Gopalakrishnan 7/28

Example: Helmholtz equation Helmholtz equation: − ∆ φ − ω 2 φ = ˆ in Ω ⊆ R N . ıω f , First order reformulation: ıω v + grad φ = � ˆ 0 , in Ω ˆ ıωφ + div v = f , in Ω. Jay Gopalakrishnan 8/28

Example: Helmholtz equation Helmholtz equation: − ∆ φ − ω 2 φ = ˆ in Ω ⊆ R N . ıω f , First order reformulation: ıω v + grad φ = � ˆ 0 , in Ω ˆ ıωφ + div v = f , in Ω. Fit to the general structure: � 0 � v � � ˆ � � v � � � v � ıω v + grad φ grad Au = A = = ˆ ıω + φ ˆ ıωφ + div v φ div 0 φ �� 0 �� 0 � � � � e 1 e N = ∂ α ( a ij α u j ) = ˆ ıω u + ∂ 1 u + · · · + ∂ N u e t e t 0 0 1 N Jay Gopalakrishnan 8/28

The Adjoint The adjoint of A is a closed (unbounded) operator A ∗ : L 2 ( Ω ) l → L 2 ( Ω ) m whose domain is � dom A ∗ = ∃ ℓ ∈ L 2 ( Ω ) m such that s ∈ L 2 ( Ω ) l : � ( Av , s ) Ω = ( v , ℓ ) Ω ∀ v ∈ dom ( A ) , and satisfies ∀ v ∈ dom A , v ∗ ∈ dom A ∗ . ( Av , v ∗ ) Ω = ( v , A ∗ v ∗ ) Ω , Assume that A ∗ u is a distribution for all u ∈ L 2 ( Ω ) l . Jay Gopalakrishnan 9/28

Graph spaces These are the graph spaces of A and A ∗ : W ( Ω ) = { u ∈ L 2 ( Ω ) m : Au ∈ L 2 ( Ω ) l } , W ∗ ( Ω ) = { u ∈ L 2 ( Ω ) l : A ∗ u ∈ L 2 ( Ω ) m } When solving a PDE Au = f in the graph space, we incorporate boundary conditions in dom A . V = dom A in W ( Ω )-topology , V ∗ = dom A ∗ in W ∗ ( Ω )-topology . Elements of the modern theory of Friedrichs systems [ Ern+Guermond+Caplain 2007 ] can be generalized to this setting. Jay Gopalakrishnan 10/28

Boundary Operator Define D : W ( Ω ) → W ∗ ( Ω ) ′ and D ∗ : W ∗ ( Ω ) → W ( Ω ) ′ by � Dw , w ∗ � W ∗ ( Ω ) = ( Aw , w ∗ ) Ω − ( w , A ∗ w ∗ ) Ω , � D ∗ w ∗ , w � W ( Ω ) = ( A ∗ w ∗ , w ) Ω − ( w ∗ , Aw ) Ω , for all w ∈ W ( Ω ) and w ∗ ∈ W ∗ ( Ω ). Lemma [DA] (Domain of the Adjoint) V ∗ = ⊥ D ( V ) = { w ∗ ∈ W ∗ ( Ω ) : � Dv , w ∗ � = 0 ∀ v ∈ V } . Jay Gopalakrishnan 11/28

Example: Helmholtz equation � v � � ˆ � ıω v + grad φ Recall the operator: Au = A = ˆ ıωφ + div v φ Add impedance boundary condition: v · n − φ = 0 , on ∂Ω. Incorporating the boundary condition into domain, � � v � � � ∈ H (div , Ω ) × H 1 ( Ω ) : dom A = V = ( v . n − φ ) ∂Ω = 0 . � φ Jay Gopalakrishnan 12/28

Example: Helmholtz equation � v � � ˆ � ıω v + grad φ Recall the operator: Au = A = ˆ ıωφ + div v φ Add impedance boundary condition: v · n − φ = 0 , on ∂Ω. Incorporating the boundary condition into domain, � � v � � � ∈ H (div , Ω ) × H 1 ( Ω ) : dom A = V = ( v . n − φ ) ∂Ω = 0 . � φ What is D? For smooth functions w , v , ψ, φ , � w � � v � � w · n ¯ � D � = φ + ψ v · n . , ψ φ ∂Ω Jay Gopalakrishnan 12/28

Example: Helmholtz equation � v � � ˆ � ıω v + grad φ Recall the operator: Au = A = ˆ ıωφ + div v φ Add impedance boundary condition: v · n − φ = 0 , on ∂Ω. Incorporating the boundary condition into domain, � � v � � � ∈ H (div , Ω ) × H 1 ( Ω ) : dom A = V = ( v . n − φ ) ∂Ω = 0 . � φ What is D? For smooth functions w , v , ψ, φ , � w � � v � � w · n ¯ � D � = φ + ψ v · n . , ψ φ ∂Ω What is dom A ∗ , or V ∗ ? Lemma [DA] = ⇒ � � w � � V ∗ = ⊥ D ( V ) = � ∈ H (div , Ω ) × H 1 ( Ω ) : ( w . n + ψ ) ∂Ω = 0 . � ψ Jay Gopalakrishnan 12/28

Next Quick introduction to DPG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Unbounded operator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Broken weak form Heat equation vs. Schr¨ odinger equation A density result DPG method for the Schr¨ odinger equation Jay Gopalakrishnan 13/28

Strong & Weak forms Classical operator form Given f ∈ L 2 ( Ω ) l , find u ∈ V satisfying Au = f . Strong Petrov-Galerkin form Find u ∈ V satisfying ∀ v ∈ L 2 ( Ω ) l . ( Au , v ) Ω = ( f , v ) Weak Petrov-Galerkin form (Unbroken) Find u ∈ L 2 ( Ω ) m satisfying ( u , A ∗ v ) Ω = ( f , v ) Ω ∀ v ∈ V ∗ . ⇒ ( u , A ∗ v ) Ω + Au = f = � Du , v � = ( f , v ) Ω � �� � Lemma [DA] = ⇒ 0 Jay Gopalakrishnan 14/28

Strong & Weak forms Classical operator form Given f ∈ L 2 ( Ω ) l , find u ∈ V satisfying Au = f . Strong Petrov-Galerkin form Find u ∈ V satisfying ∀ v ∈ L 2 ( Ω ) l . ( Au , v ) Ω = ( f , v ) Weak Petrov-Galerkin form (Unbroken) Find u ∈ L 2 ( Ω ) m satisfying ( u , A ∗ v ) Ω = ( f , v ) Ω ∀ v ∈ V ∗ . This form is not amenable to DPG discretization! Jay Gopalakrishnan 14/28