On the global convergence of a singularly perturbed parabolic - PowerPoint PPT Presentation

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh On the global convergence of a singularly perturbed parabolic problem of reaction diffusion type with a discontinuous initial condition J.L. Gracia, E. O’Riordan Department of Applied Mathematics School of Mathematics University of Zaragoza (Spain) Dublin City University (Ireland) Workshop “Numerical Analysis for singularly perturbed Problems” Dedicated to the 60th birthday of Martin Stynes 16th–18th November 2011, TU Dresden Research supported by the project MEC/FEDER MTM 2010-16917 and the Diputaci´ on General de Arag´ on

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Martin Happy 60th!!!!

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh On the global convergence of a singularly perturbed parabolic problem of reaction diffusion type with a discontinuous initial condition J.L. Gracia, E. O’Riordan Department of Applied Mathematics School of Mathematics University of Zaragoza (Spain) Dublin City University (Ireland) Workshop “Numerical Analysis for singularly perturbed Problems” Dedicated to the 60th birthday of Martin Stynes 16th–18th November 2011, TU Dresden Research supported by the project MEC/FEDER MTM 2010-16917 and the Diputaci´ on General de Arag´ on

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Outline Talk is split into two parts. 1. Fitted mesh method for reaction-diffusion problem with smooth initial condition, but containing a layer in the initial condition. 2. Reaction-diffusion problem with discontinuous data. 1 Fitted operator method (with uniform mesh) when initial condition is discontinuous 2 Approximate problem by regularizing initial condition 3 Fitted mesh method (with classical operator) 4 Fitted operator combined with fitted mesh method

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Uniformly convergent schemes We are interesting in robust schemes (or uniformly convergent schemes) 0 <ε ≤ 1 � U N ε − u ε � ∞ ≤ CN − p , max p > 0 where C is a constant independent of the discretization parameters N and (also) the singular perturbation parameters ε . A priori numerical methods: Fitted operator methods (Il’in, El-Mistikawya and Werle, ...) (Quasi–)Uniform mesh + special discrete operator Fitted mesh methods Layer–adapted meshes (Bakhvalov, Shishkin, ...) + classical discrete operator

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Parabolic reaction-diffusion problem I L ε u := − ε u xx + b ( x , t ) u + p ( x , t ) u t = f ( x , t ) , ( x , t ) ∈ (0 , 1) × (0 , T ] , u ( x , 0) = φ ( x ; ε, d ) , 0 ≤ x ≤ 1 , u (0 , t ) = 0 , u (1 , t ) = 1 , 0 < t ≤ T , b ( x , t ) ≥ β > 0 , p ( x , t ) ≥ p 0 > 0 , where the initial condition φ is smooth, but is constructed to have an artificial interior layer in the vicinity of a point x = d , 0 < d < 1 and C √ ε ≤ d ≤ 1 − δ < 1. if x ∈ Ω − := (0 , d ) ,  0 ,  φ ( x ; ε, d ) = � ε ( x − d ) � p β 1 � if x ∈ Ω + := ( d , 1) , 1 − e − K ,  where p ≥ 4, β 1 ≥ 2 β > 0 , K are such that φ (1; ε, d ) = 1 . Throughout Q − := Ω − × (0 , T ] , Q + := Ω + × (0 , T ] .

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Solution decomposition Maximum Principle The differential operator satisfies a maximum principle, and from this we deduce that � u � ¯ Q ≤ C , where � u � ¯ Q = max ( x , t ) ∈ ¯ Q | u ( x , t ) | is the maximum norm. To identify the layer structure of the solution, we consider a decomposition of the solution of the form u = v + w + z , into a discontinuous regular component v , a continuous boundary layer component w and a discontinuous interior layer component z .

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Theorem There exists functions r 0 ( t ) , r 1 ( t ) , r 2 ( t ) such that the solutions v − , v + of the problems L ε v − = f , ( x , t ) ∈ Q − , v − (0 , t ) = 0 , v − ( x , 0) = 0 , L ε v + = f , ( x , t ) ∈ Q + , v + ( x , 0) = 1 , v − ( d , t ) = r 0 ( t ) , v + ( d , t ) = r 1 ( t ) , v + (1 , t ) = r 2 ( t ) , satisfy v − ∈ C 4+ γ ( ¯ Q − ) , v + ∈ C 4+ γ ( ¯ Q + ) and for 0 ≤ j + 2 m ≤ 4 the bounds � ∂ j + m v − � � C (1 + ε 1 − j / 2 ) , ≤ � � � ¯ ∂ x j ∂ t m Q − � ∂ j + m v + � � C (1 + ε 1 − j / 2 ) . ≤ � � � ¯ ∂ x j ∂ t m Q +

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Decompose u − v into two subcomponents u − v = w + z The boundary layer function w is the solution of w ≡ 0 , ( x , t ) ∈ ¯ Q − ; L ε w = 0 , ( x , t ) ∈ Q + ; w ( d , t ) = 0 , w ( x , 0) ≈ 0 , w (1 , t ) = u (1 , t ) − v + (1 , t ) , Theorem The boundary layer function w ∈ C 0 ( ¯ Q ) ∩ C 4+ γ ( ¯ Q + ) and � ∂ j + m w � β � � ε (1 − x ) , 0 ≤ j +2 m ≤ 4 , ( x , t ) ∈ Q + . � ≤ C ε − j / 2 e − ∂ x j ∂ t m ( x , t ) � �

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh The discontinuous multi-valued interior layer function z satisfies L ε z = 0 , ( x , t ) ∈ Q − ∪ Q + , [ z ]( d , t ) = − [ v ]( d , t ) , [ z x ]( d , t ) = − [ v x + w x ]( d , t ) , t ≥ 0 , z (0 , t ) = z (1 , t ) = 0 , t ≥ 0 , z ( x , 0) = 0 , x ∈ Ω − , � β z ( x , 0) ≈ e − ε ( x − d ) , x ∈ Ω + . Theorem Assume C √ ε ≤ d ≤ 1 − δ < 1 . The interior layer function z ∈ C 4+ γ ( ¯ Q − ) ∪ C 4+ γ ( ¯ Q + ) and for 0 ≤ j + 2 m ≤ 4 � ∂ j + m z � � � β C ε − j / 2 e − ε ( d − x ) , ( x , t ) ∈ Q − , ∂ x j ∂ t m ( x , t ) ≤ � � � � ∂ j + m z � β � � ε ( x − d ) , ( x , t ) ∈ Q + . C ε − j / 2 e − ∂ x j ∂ t m ( x , t ) ≤ � � �

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Numerical method x U N , M + b ( x i , t j ) U N , M + p ( x i , t j ) D − t U N , M = f ( x i , t j ) , t j > 0 , − εδ 2 U N , M ( x i , 0) = φ ( x i ; ε, d ) , 0 < x i < 1 , U N , M (0 , t j ) = 0 , U N , M (1 , t j ) = 1 , t j ≥ 0 . Split the space domain using a Shishkin mesh [0 , d − τ 1 ] ∪ [ d − τ 1 , d + τ 2 ] ∪ [ d + τ 2 , 1 − τ 2 ] ∪ [1 − τ 2 , 1] , with � ε � ε τ 1 = min { d β ln N } , τ 2 = min { 1 − d 2 , 2 , 2 β ln N } . 4 INTERIOR LAYER BOUNDARY LAYER

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Theorem (G., O’Riordan, submitted) U N be the Let u be the solution of the continuous problem and let ¯ linear interpolant of the solution of the discrete problem. Then, U N , M − u � ¯ Q ≤ C (( N − 1 ln N ) 2 + N − 1 ε 1 / 2 + ∆ t ) . � ¯ Limiting case of d = 0 Consider the following initial condition � ε x � p β 1 � 1 − e − φ ( x ; ε ) = K , p ≥ 4 . The space domain split into [0 , σ c ] ∪ [ σ c , 1 − σ c ] ∪ [1 − σ c , 1] , where � ε σ c := min { 1 4 , 2 β ln N } . The error bound given above still applies.

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Test problem − ε u xx + u + u t = 4 x (1 − x ) e t , ( x , t ) ∈ (0 , 1) × (0 , 0 . 5] , u ( x , 0) = φ ( x ; ε, d ) , 0 < x ≤ 1 , u (0 , t ) = 0 , u (1 , t ) = 1 , 0 < t ≤ 1 , "Solution for eps=1.d-6" 1.2 1 0.8 0.6 0.4 0.2 0 0.5 1 0.45 0.4 0.8 0.35 0.3 0.6 0.25 0.2 0.4 0.15 0.1 0.2 0.05 0 0 Figure: Numerical approximation for ε = 10 − 6 and d = 0.

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Parabolic reaction-diffusion problem II Consider the test problem involving the heat equation u t − ε u xx = 0 , ( x , t ) ∈ (0 , 1) × (0 , 1] , and the initial and boundary conditions are taken such that u ( x , t ) = w 0 ( x , t ) = 1 x 2 erf ( 2 √ ε t ) , with � ζ 2 exp( − α 2 ) d α erf ( ζ ) = √ π 0 is the error function. The initial condition is discontinuous at (0,0), which generates a boundary/corner layer

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Test problem 0.5 0.4 0.3 0.2 0.1 0 1 0.8 1 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 Figure: Exact solution for ε = 2 − 10 .

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh Numerical approximation: We will numerically examine the performance of five approximation methods. 1 Classical scheme on uniform mesh 2 Fitted operator [Hemker and Shishkin, 1994] on uniform mesh 3 Classical operator on fitted mesh applied to regularized problem. 4 Classical operator on fitted mesh. 5 Fitted operator on fitted mesh.