Weak Galerkin Finite Element Methods for Elliptic and Parabolic Problems on Polygonal Meshes MWNDEA 2020 Naresh Kumar Department of Mathematics Indian Institute of Technology Guwahati, Guwahati, India Naresh Kumar (IITG) Weak Galerkin FEM 1 / 32
Outline • Weak Galerkin Finite Element Methods. • Motivation • Implementation • WG-FEM for Model PDEs. • Second Order Elliptic Problems. • Second Order Parabolic Problems. • Numerical Results. Joint Work With: Prof. Bhupen Deka. Naresh Kumar (IITG) Weak Galerkin FEM 2 / 32
Basic Notation • We denote by H m ( J ; B ), 1 ≤ m < ∞ , the space of all measurable functions φ : J → B for which � 1 � T � m 2 2 � ∂ j u ( t ) � � � � � u � H m ( J ; B ) = dt < ∞ . � � ∂ t j � � 0 B j =0 • The minimal regularity space X = L 2 (0 , T ; H m +1 (Ω)) ∩ H 1 (0 , T ; H m − 1 (Ω)) , 0 equipped with the norm � v � 2 X = � v � 2 L 2 (0 , T ; H m +1 (Ω)) + � ∂ t v � 2 L 2 (0 , T ; H m − 1 (Ω)) . • We will use � · � for L 2 -norm. Naresh Kumar (IITG) Weak Galerkin FEM 3 / 32
Second Order Elliptic Problems Find u ∈ H 1 0 (Ω) such that ( a ∇ u , ∇ v ) = ( f , v ) ∀ v ∈ H 1 0 (Ω) . Procedures in the standard Galerkin finite element method: • Partition Ω into triangles or tetrahedra. • Construct a subspace, denoted by S h ⊂ H 1 0 (Ω), using piecewise polynomials. • Seek for a finite element solution u h from S h such that ( a ∇ u h , ∇ v ) = ( f , v ) ∀ v ∈ S h . The classical gradient ∇ u for u ∈ C 1 (Ω) can be computed as: � � � u ( φ. n ) ∀ φ ∈ [ C 1 (Ω)] 2 ∇ u .φ = − u ∇ .φ + K K ∂ K Thus, u can be extended to { u 0 , u b } with ∇ u being extended to ∇ w u . Naresh Kumar (IITG) Weak Galerkin FEM 4 / 32
Motivation: Weak Function • Let K be any polygonal or polyhedral domain with interior K 0 and boundary ∂ K . • A weak function on the region K refers to a pair of scalar-valued functions 1 v = { v 0 , v b } such that v 0 ∈ L 2 ( K ) and v b ∈ H 2 ( ∂ K ). • Denote by V ( K ) the space of weak functions on K ; i. e. , 1 V ( K ) = { v = { v 0 , v b } : v 0 ∈ L 2 ( K ) , v b ∈ H 2 ( ∂ K ) } . • For any weak function v = { v 0 , v b } , its weak gradient ∇ w v is defined (interpreted) as a linear functional on H (div , K ) whose action on each q ∈ H (div , K ) is given by � � � ∇ w v . qdK = − v 0 ∇ · qdK + v b q · n ds , (1) K K ∂ K where n is the outward normal to ∂ K . Naresh Kumar (IITG) Weak Galerkin FEM 5 / 32
Inclusion Result • The Sobolev space H 1 ( K ) can be embedded into the space V ( K ) by an inclusion map i V : H 1 ( K ) �→ V ( K ) defined as follows i V ( φ ) = { φ | K , φ | ∂ K } , φ ∈ H 1 ( K ) . • With the help of the inclusion map i V , the Sobolev space H 1 ( K ) can be viewed as a subspace of V ( K ) by identifying each φ ∈ H 1 ( K ) with i V ( φ ). • Analogously, a weak function v = { v 0 , v b } ∈ V ( K ) is said to be in H 1 ( K ) if it can be identified with a function φ ∈ H 1 ( K ) through the above inclusion map. • For u ∈ H 1 ( K ), we have i V ( u ) = { u | K , u | ∂ K } . It is not hard to see that the weak gradient is identical with the strong/classical gradient. Naresh Kumar (IITG) Weak Galerkin FEM 6 / 32
Discrete Weak Gradient Discrete Weak Gradient Operator: A discrete weak gradient operator, denoted by ∇ w , m , is defined as the unique polynomial ( ∇ w , m v ) ∈ [ P m ( K )] 2 that satisfies the following equation � � � v b ( φ · n ) ds ∀ φ ∈ [ P m ( K )] 2 , ∇ w , m v .φ dK = − v 0 ( ∇ · φ ) dK + (2) K K ∂ K 1 where v = { v 0 , v b } such that v 0 ∈ L 2 ( K ) and v b ∈ H 2 ( ∂ K ). Naresh Kumar (IITG) Weak Galerkin FEM 7 / 32
Weak Galerkin Space � 2 ) � Weak Galerkin space is defined as: ( P k ( K ) , P j ( ∂ K ) , P l ( K ) • k ≥ 1 is the degree of polynomials in the interior of the element K , • j ≥ 0 is the degree of polynomials on the boundary of K and • l ≥ 0 is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. • k , j , l are selected in such a way that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. Lowest Order Weak Galerkin Space � 2 ). • A lowest order WG-FEM space is ( P 1 ( K ) , P 0 ( ∂ K ) , � P 0 ( K ) Naresh Kumar (IITG) Weak Galerkin FEM 8 / 32
Weak Galerkin Approximation � 2 ) . � We choose weak Galerkin space is ( P k ( K ) , P k ( ∂ K ) , P k − 1 ( K ) For k ≥ 1, let V h be WG FE space associated with T h & defined as: V h = { v = { v 0 , v b } : v 0 | K 0 ∈ P k ( K ) , v b | e ∈ P k ( e ) , e ∈ ∂ K , K ∈ T h } . Naresh Kumar (IITG) Weak Galerkin FEM 9 / 32
Weak Galerkin Approximation Cont... • Note that functions in V h are defined on each element, and there are two-sided values of v b on each interior edge/face e , depicted as v b | ∂ T 1 and v b | ∂ T 2 in above Figure. • We assume that v b has unique value on each interior edge/face e that is ∀ e ∈ E 0 [ v ] e = 0 h , [ v ] e denotes the jump of v ∈ V h across an interior edge e ∈ E 0 h . • We write V 0 h = { v = { v 0 , v b } ∈ V h : v b = 0 on ∂ Ω } . • For v ∈ V h , the discrete weak gradient of it is defined as the unique polynomial ( ∇ w v ) ∈ [ P k − 1 ( K )] 2 that satisfies the following equation � � � v b ( φ · n ) ds ∀ φ ∈ [ P k − 1 ( K )] 2 . (3) ∇ w , m v .φ dK = − v 0 ( ∇· φ ) dK + K K ∂ K Naresh Kumar (IITG) Weak Galerkin FEM 10 / 32
Shape Regularity for Polytopal Elements Why Shape Regularity? The shape regularity is needed for 1 trace inequality. 2 inverse inequality. 3 domain inverse inequality. ( A weak Galerkin mixed finite element method for second order elliptic problems , Math. Comp., 83 (2014), 2101-2126, by J. Wang and X. Ye for more details) Naresh Kumar (IITG) Weak Galerkin FEM 11 / 32
Elliptic Boundary Value Problem Consider following BVP � −∇ · ∇ u ) = f in Ω , (4) with boundary condition u = 0 on ∂ Ω . (5) Weak Formulation: Find u ∈ H 1 0 (Ω) such that � � ∇ u · ∇ vdx = fvdx . (6) Ω Ω Naresh Kumar (IITG) Weak Galerkin FEM 12 / 32
Elliptic Boundary Value Problem Cont... Weak Galerkin Approximation : Find u h = { u 0 , u b } ∈ V 0 h such that ∀ v h ∈ V 0 a s ( u h , v h ) = ( f , v h ) h , (7) with a s ( u h , v h ) = a ( u h , v h ) + s ( u h , v h ) , (8) where • a ( · , · ) : V h × V h → R is a bilinear map given by � a ( u h , v h ) = ( ∇ w u h , ∇ w v h ) = ( ∇ w u h , ∇ w v h ) K , (9) K ∈T h • with a stabilizer s ( · , · ) : V h × V h → R defined by � h − 1 s ( u h , v h ) = K � u 0 − u b , v 0 − v b � ∂ K . (10) K ∈T h • It is important to check that a s ( · , · ) is positive so that WG approximation (7) has a unique solution. In fact bilinear map a s ( · , · ) induces a norm. Naresh Kumar (IITG) Weak Galerkin FEM 13 / 32
Discrete Norms • We consider following norm associated with the bilinear map a s ( · , · ) � ||| u h ||| = a s ( u h , u h ) . (11) • For simplicity, we shall only verify the positive length property for |||·||| . • Assume that ||| w ||| = 0 for some w = { w 0 , w b } ∈ V 0 h . • It follows that ∇ w w = 0 on each element K ∈ T h and w 0 = w b on ∂ K . • Thus, we have from the definition of weak gradient that for any φ ∈ [ P k − 1 ( K )] 2 . 0 = ( ∇ w w , φ ) K = − ( w 0 , ∇ · φ ) K + � w b , φ · n � ∂ K = ( ∇ w 0 , φ ) + � w b − w 0 , φ · n � ∂ K = ( ∇ w 0 , φ ) K . • Letting φ = ∇ w 0 in the above equation yields ∇ w 0 = 0 on K ∈ T h . Naresh Kumar (IITG) Weak Galerkin FEM 14 / 32
Discrete Norms Contd. • It follows that w 0 = constant on every K ∈ T h . • This, together with the fact that w b = w 0 on ∂ K and w b = 0 on ∂ Ω, implies that w 0 = 0 and w b = 0. • We define following discrete H 1 -norm � 1 2 , v = { v 0 , v b } ∈ V 0 � � ( �∇ v 0 � 2 K + h − 1 K � v 0 − v b � 2 � v � 1 , h = ∂ K ) h . K ∈T h • The following lemma indicates that discrete H 1 -norm is equivalent to triple bar norm ( c.f. Lemma 5.3, A weak Galerkin finite element method with polynomial reduction, JCAM, 285 (2015) 4558) Lemma There exist two positive constants C 1 and C 2 such that C 1 � v � 1 , h ≤ ||| v ||| ≤ C 2 � v � 2 , h ∀ v h ∈ V h . � Naresh Kumar (IITG) Weak Galerkin FEM 15 / 32
Recommend
More recommend
