A Domain Decomposition Method for Quasilinear Elliptic PDEs Using - PowerPoint PPT Presentation

W I S S E N T E C H N I K L E I D E N S C H A F T A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements Matthias Gsell and Olaf Steinbach Institute of Computational Mathematics DD23, International Conference on Domain Decomposition Methods www.numerik.math.tugraz.at

www.numerik.math.tugraz.at Outline 1. Motivation 2. Approach – Continuous Formulation – Discretization Strategies 3. Numerical Example 4. Outlook Matthias Gsell, Institute of Computational Mathematics 2 2015-07-06

www.numerik.math.tugraz.at Outline 1. Motivation 2. Approach – Continuous Formulation – Discretization Strategies 3. Numerical Example 4. Outlook Matthias Gsell, Institute of Computational Mathematics 3 2015-07-06

www.numerik.math.tugraz.at Motivation See [Berningner, 2008]. To describe the flow of fluid (water) in porous media, we use the Richards equation for the physical pressure p . Richards Equation � �� � �� � � n ( x ) ∂θ ( p ( x , t )) K ( x ) − ∇ · ∇ p ( x , t ) − ̺ g z k r θ p ( x , t ) = f ( x , t ) ∂ t µ Quantities θ . . . saturation n . . . porosity ̺ . . . density µ . . . viscosity Ω K . . . permeability k r . . . relative permeability g . . . gravitational const. f . . . source term Γ Matthias Gsell, Institute of Computational Mathematics 4 2015-07-06

www.numerik.math.tugraz.at Motivation Consider the heat equation to describe the distribution of heat where the thermal conductivity depends on the heat p . Heat Equation � � � � ∂ p ( x , t ) − ∇ · ∇ p ( x , t ) k p ( x , t ) = f ( x , t ) ∂ t Quantities k . . . thermal conductivity f . . . heat source Ω Γ Matthias Gsell, Institute of Computational Mathematics 5 2015-07-06

www.numerik.math.tugraz.at Motivation Consider the heat equation to describe the distribution of heat where the thermal conductivity depends on the heat p . Heat Equation � � � � ∂ p ( x , t ) − ∇ · ∇ p ( x , t ) k p ( x , t ) = f ( x , t ) ∂ t Quantities k . . . thermal conductivity f . . . heat source Ω Γ � � � � Both equtions have same second order term −∇ · k p ( x , t ) ∇ p ( x , t ) . Matthias Gsell, Institute of Computational Mathematics 5 2015-07-06

www.numerik.math.tugraz.at Motivation Let Ω ⊂ R d . For given data f , g N , g D consider the following quasilinear boundary value problem. Model Problem Find p , such that � � � � −∇ · k p ( x ) ∇ p ( x ) = f ( x ) in Ω p ( x ) = g D ( x ) on Γ D � � k p ( x ) ∇ p ( x ) · n Γ N = g N ( x ) on Γ N Matthias Gsell, Institute of Computational Mathematics 6 2015-07-06

www.numerik.math.tugraz.at Motivation Let Ω ⊂ R d . For given data f , g N , g D consider the following quasilinear boundary value problem. Model Problem Find p , such that � � � � −∇ · k p ( x ) ∇ p ( x ) = f ( x ) in Ω p ( x ) = g D ( x ) on Γ D � � k p ( x ) ∇ p ( x ) · n Γ N = g N ( x ) on Γ N Assumptions on k : R → R . • ) k ∈ L ∞ ( R ) , i.e. | k ( s ) | ≤ C < ∞ for almost all s ∈ R . • ) there exists a constant c > 0 such that 0 < c ≤ k ( s ) for almost all s ∈ R . • ) k is Lipschitz continuous, i.e. | k ( s ) − k ( t ) | ≤ L | s − t | . Matthias Gsell, Institute of Computational Mathematics 6 2015-07-06

www.numerik.math.tugraz.at Motivation Consider the weak formulation of the BVP . Weak Formulation Find p ∈ H 1 g D , Γ D (Ω) , such that � � � k ( p ) ∇ p · ∇ v d x = f v d x + g N v d s x Ω Ω Γ N is satisfied for all v ∈ H 1 0 , Γ D (Ω) . Matthias Gsell, Institute of Computational Mathematics 7 2015-07-06

www.numerik.math.tugraz.at Motivation Consider the weak formulation of the BVP . Weak Formulation Find p ∈ H 1 g D , Γ D (Ω) , such that � � � k ( p ) ∇ p · ∇ v d x = f v d x + g N v d s x Ω Ω Γ N is satisfied for all v ∈ H 1 0 , Γ D (Ω) . We can apply the Kirchhoff transformation to the physical quantity p and introduce the generalized quantity u as � p ( x ) � � u ( x ) := κ p ( x ) = k ( s ) d s . 0 Matthias Gsell, Institute of Computational Mathematics 7 2015-07-06

www.numerik.math.tugraz.at Motivation Consider the weak formulation of the BVP . Weak Formulation Find p ∈ H 1 g D , Γ D (Ω) , such that � � � k ( p ) ∇ p · ∇ v d x = f v d x + g N v d s x Ω Ω Γ N is satisfied for all v ∈ H 1 0 , Γ D (Ω) . We can apply the Kirchhoff transformation to the physical quantity p and introduce the generalized quantity u as � p ( x ) � � u ( x ) := κ p ( x ) = k ( s ) d s . 0 Therefore we get ∇ u ( x ) = κ ′ � � � � p ( x ) ∇ p ( x ) = k p ( x ) ∇ p ( x ) . Matthias Gsell, Institute of Computational Mathematics 7 2015-07-06

www.numerik.math.tugraz.at Motivation Transformed Model Problem Find u ∈ H 1 h D , Γ D (Ω) , such that � � � ∇ u · ∇ v d x = f v d x + g N v d s x Ω Ω Γ N is satisfied for all v ∈ H 1 0 , Γ D (Ω) with h D := κ ( g D ) . • ) Unique solvability from Lax–Milgram Lemma. • ) Numerical analysis is well known for this problem. • ) Apply inverse Kirchhoff transformation to obtain the physical quantity p . Matthias Gsell, Institute of Computational Mathematics 8 2015-07-06

www.numerik.math.tugraz.at Motivation Transformed Model Problem Find u ∈ H 1 h D , Γ D (Ω) , such that � � � ∇ u · ∇ v d x = f v d x + g N v d s x Ω Ω Γ N is satisfied for all v ∈ H 1 0 , Γ D (Ω) with h D := κ ( g D ) . • ) Unique solvability from Lax–Milgram Lemma. • ) Numerical analysis is well known for this problem. • ) Apply inverse Kirchhoff transformation to obtain the physical quantity p . ⇒ We considered nonlinearities of the form k : R → R . Matthias Gsell, Institute of Computational Mathematics 8 2015-07-06

www.numerik.math.tugraz.at Motivation Transformed Model Problem Find u ∈ H 1 h D , Γ D (Ω) , such that � � � ∇ u · ∇ v d x = f v d x + g N v d s x Ω Ω Γ N is satisfied for all v ∈ H 1 0 , Γ D (Ω) with h D := κ ( g D ) . • ) Unique solvability from Lax–Milgram Lemma. • ) Numerical analysis is well known for this problem. • ) Apply inverse Kirchhoff transformation to obtain the physical quantity p . ⇒ We considered nonlinearities of the form k : R → R . ⇒ Try nonlinearities of the form k : R × Ω → R . Matthias Gsell, Institute of Computational Mathematics 8 2015-07-06

www.numerik.math.tugraz.at Outline 1. Motivation 2. Approach – Continuous Formulation – Discretization Strategies 3. Numerical Example 4. Outlook Matthias Gsell, Institute of Computational Mathematics 9 2015-07-06

www.numerik.math.tugraz.at Approach Continuous Formulation � � � � Consider a second order term is of the form −∇ · k p ( x ) , x ∇ p ( x ) . Richards equation Heat equation Γ 2 Ω 2 Γ Ω 2 Γ Ω 1 Ω 1 Γ 1 Γ 1 Different material types. Different soil parameter. ⇒ Different conductivities. ⇒ Different permeabilities. Matthias Gsell, Institute of Computational Mathematics 10 2015-07-06

www.numerik.math.tugraz.at Approach Continuous Formulation � � � � Consider a second order term is of the form −∇ · k p ( x ) , x ∇ p ( x ) . Richards equation Heat equation Γ 2 Ω 2 Γ Ω 2 Γ Ω 1 Ω 1 Γ 1 Γ 1 Different material types. Different soil parameter. ⇒ Different conductivities. ⇒ Different permeabilities. • ) Apply Kirchhoff transformation. p ( x ) � � � � � ⇒ u ( x ) := κ p ( x ) = k ( s , x ) d s but ∇ u ( x ) � = k p ( x ) , x ∇ p ( x ) . 0 Matthias Gsell, Institute of Computational Mathematics 10 2015-07-06

www.numerik.math.tugraz.at Approach Continuous Formulation � � � � Consider a second order term is of the form −∇ · k p ( x ) , x ∇ p ( x ) . Richards equation Heat equation Γ 2 Ω 2 Γ Ω 2 Γ Ω 1 Ω 1 Γ 1 Γ 1 Different material types. Different soil parameter. ⇒ Different conductivities. ⇒ Different permeabilities. • ) Apply Kirchhoff transformation. p ( x ) � � � � � ⇒ u ( x ) := κ p ( x ) = k ( s , x ) d s but ∇ u ( x ) � = k p ( x ) , x ∇ p ( x ) . 0 ⇒ new approach to exploit the advantages of the Kirchhoff transformation. Matthias Gsell, Institute of Computational Mathematics 10 2015-07-06

www.numerik.math.tugraz.at Approach Continuous Formulation Let Ω ⊂ R d . For given data f , g N , g D consider the following quasilinear boundary value problem. Model Problem Find p , such that � � −∇ · k ( p ( x ) , x ) ∇ p ( x ) = f ( x ) in Ω p ( x ) = g D ( x ) on Γ D k ( p ( x ) , x ) ∇ p ( x ) · n Γ N = g N ( x ) on Γ N Γ D Assumption : Ω 4 � � � K � � Ω 3 • ) k p ( x ) , x = X Ω i ( x ) k i p ( x ) Ω 5 i = 1 Ω 1 • ) k i ∈ L ∞ ( R ) and 0 < c i ≤ k i for c i ∈ R , Ω 2 • ) k i is Lipschitz continuous. Γ N Matthias Gsell, Institute of Computational Mathematics 11 2015-07-06