Weak Galerkin Finite Element Scheme and Its Applications

Weak Galerkin Finite Element Scheme and Its Applications Ran Zhang Department of Mathematics Jilin University, China IMS, Singapore February 6, 2015

Talk Outline Motivation WG FEMs: Weak Operators + Stabilizer Weak operators: Generalized Weak Derivatives Stabilizer: adding some weak continuity Applications on WG methods Summary

I. Motivation

Second order elliptic equation Consider second order elliptic problem: Testing (1) by v H0 1 (Ω) gives uvdx = u vdx that is, Ω u = f, in Ω (1) Ω u = 0, on Ω. (2) Ω ( u, v) = (f, v). u n vds = fvdx, Ω

Classical Galerkin FEMs V h H 1 0 (Ω) Weak form: find u H0 1 (Ω) such that ( u, v) = (f, v), v H 1 0(Ω). (3) Partition Ω into triangles or tetrahedra T h, let V h H0 1 (Ω) be a finite dimensional space. V h = {v H 1 0 (Ω); v T P k (T), T T h }. Continuous finite element method: find u h V h such that ( u h, v h ) = (f,v h ), v V h, (4)

Bases for Conforming FEM

Some Features of the Classical Galerkin FEM V h is a subspace of the space from where the exact solution was sought. V h must have good approximation properties for small h. Functions in V h are defined in classical sense. The gradient φ is computed in the classical sense ( u h, v) = (f,v).

Classical Non-conforming Galerkin FEM V h H 1 0 (Ω) P1 non-conforming FEM: piecewise linear functions which are continuous at the mid-point of each edge. Morley element and Wilson brick: piecewise polynomials with partial continuity across each edge/face. Makes use of the same variational form. Derivatives are taken element-by-element: ( u h, v) T = (f,v). T

Some Features of the Classical Non-conforming FEM V h is not a subspace of the space from where the exact solution was sought, but is close to be a subspace. V h must have good approximation properties for small h. Functions in V h are defined in classical sense, though not continuous. Function plug-in : the gradient φ is computed element by element in the classical sense.

A Formal Change of the Standard Galerkin Formally, we may replace v by any distribution, and v by another distribution, say w v, and seek for a distribution u h such that Critical Issues: ( w u h, w v) = (f,v) v V h. Why do we introduce the operator w? Functions in V h are allowed to be more general (as distributions or generalized functions) in what format? The gradient v is computed weakly, also as distributions how? Convergence and accuracy of the resulting schemes? Robustness?

Distributions and L 1 loc (Ω) For any open domain Ω R d D(Ω) = C0 (Ω): space of test functions with compact support D (Ω): the dual of D(Ω) consisting of continuous linear functionals, known as distributions on Ω. Any locally integrable function u L 1 loc (Ω) can be identified as a distribution by u,φ = u(x)φ(x)dx, φ D(Ω). Ω

Weak Derivatives and Sobolev Spaces For any multi-index α = (α 1,...,α d ) with α = α 1 +... + α d Definition For u L 1 loc (Ω) and any α, v is called the αth order weak derivative of u if v L 1 loc (Ω) and satisfies v,φ = ( 1) α u, α φ φ D(Ω). Sobolev space W k,p (Ω) = {u : u L 1 loc (Ω), α u L p (Ω)} equipped with the obvious norm.

Weak Derivative and Boundary For any multi-index α and u L 1 loc (Ω). α u characterizes the local property of u in the domain α u does not extend to the boundary of the domain Ω. functions in W k,p (Ω) can be extended to boundary of Ω continuously for certain combination of k,p. or equivalently, higher regularity is required if one wants to extend β u to the boundary Ω. The need of Generalized Weak Derivatives.

Motivation for... Let K R d be open bounded, and u C 1 ( K). Then, ν u(x)φ(x)dk = u(x) ν φ(x)dk+ (n ν)u(x)φ(x)ds K for all φ C 1 ( K). K The integrals on the right-hand side requires only u 0 = u in the interior of K, plus u b = u (trace) on the boundary K. We thus formally have ν u(x)φ(x)dk = u 0 (x) ν φ(x)dk+ (n ν)u b (x)φ(x)ds K K K K

Generalized Weak Derivatives Definition For any u = {u 0 ;u b } with u 0 L 2 (K) and u b L 2 ( K), the generalized weak derivative of u in the direction ν is the following linear functional on H 1 (K): ν u,φ = u 0 (x) ν φ(x)dk + (n ν)u b (x)φ(x)ds for all φ H 1 (K). K K

II. WG FEMs: Weak Operators + Stabilizer

Weak Functions For the second order elliptic problem, the space of weak functions is given by W (K) = {v = {v 0,v b } : v 0 L 2 (K), v b L 2 ( K)}. A weak function on the region K refers to a generalized function v = {v 0,v b } such that v 0 L 2 (K) and v b L 2 ( K). The first component v 0 can be understood as the value of v in the interior of K, and the second component v b is the value of v on the boundary of K. v b may not be necessarily related to the trace of v 0 on K should a trace be defined.

Illustration of the features of Weak Functions

Weak Gradient The dual of L 2 (K) can be identified with itself by using the standard L 2 inner product as the action of linear functionals. With a similar interpretation, for any v W (K), the weak gradient of v can be defined as a linear functional w v in the dual space of H 1 (K) whose action on each q [H 1 (K)] d is given by ( w v,q) := v 0 qdk + v b q nds, K K where n is the outward normal direction to K. Weak gradients become to be strong gradients if weak functions are sufficiently smooth (e.g., as restriction of smooth functions).

Discrete Weak Gradients Weak gradients have to be approximated, which leads to discrete weak gradients: w,r is given by w,r v qdk = v 0 qdk + v b q nds, K K for all q V (K,r). Here V (K,r) [P r (K)] 2. P r (K) is the set of polynomials on K with degree r 0. v = {v 0,v b } : v 0 P j (T),v b P l ( T). K

Weak Finite Element Spaces T h : partition of the domain Ω, shape regular construct local discrete elements W (T,j,l) := {v = {v 0,v b } : v 0 P j (T),v b P l ( T)}. patch local elements together to get a global space V h := {v = {v 0,v b } : {v 0,v b } T W (T,j,l), T T h }. Weak finite element spaces with homogeneous boundary value: V 0 h := {v = {v 0,v b } V h,v b T Ω = 0, T T h }.

Weak Galerkin Finite Element Formulation Weak Galerkin FEMs Algorithm I Find u h = {u 0 ;u b } V 0 h such that ( w u h, w v) = (f,v 0 ), v = {v 0 ;v b } V 0 h, where w v V (K,r) is given on each element T by w v qdt = v 0 qdt + v b q nds, T for all q V (K,r). T Robustness of V (K,r) and Vh 0? How should it be designed? T

Key Players and Their Relation There are three polynomial spaces involved in WG: P j (T): building block for v 0 on each element. P l ( T): building block for v b on the boundary of each element. P r (T): building block for the discrete weak gradient/derivative. Issues to consider: 1 What combination of j,l, and r shall produce stable finite elements?

Desired Properties for w There are two properties that are preferred for w : P1: For any v V h (j,l), if w v = 0 on T, then one must have v constant on T. In other words, v 0 = v b = constant on T; P2: Let u H m (Ω)(m 1) be a smooth function on Ω, and Q h u be a certain interpolation/projection of u in the finite element space V h (j,l). Then, w (Q h u) should be a good approximation of u.

Two WG Elements Derived from MFEM The first WG element has the following configuration: v 0 P j (T) and v b P j (e), j 0 V (T,r) = [P j (T)] 2 + xˆp j (T) (Raviart-Thomas Element) The second WG element has the following configuration: v 0 P j (T) and v b P j+1 (e), j 0 V (T,r) = [P j+1 (T)] 2 (BDM Element) Main Features Both P1 and P2 are satisfied. The gradient space uses polynomials of relatively high order. The method is limited to traditional finite element partitions of triangles or tetrahedra.

A Commutative Diagram P2 is satisfied if one has the following commutative diagram: H 1 (T) Q h S h (T) w [L 2 (T)] d Q h V (r,t) Or equivalently, the commutative property holds true w (Q h u) = Q h ( u), u H 1 (T). use low order of polynomials for the gradient space

Stabilization Term How about the Property 1? P1 does not need to be satisfied, it can be substituted through the use of stabilization. Weak Galerkin FEMs Algorithm II Seeking u h V h satisfying ( w u h, w v) + s(u h, v) = (f, v), v V h. with s(u h, v) = T T h h 1 T Q bu 0 u b,q b v 0 v b T. Key in WG: discrete weak gradient + stabilization in u to enforce weak continuities.

Shape Regularity for Elements WG with Stabilizer works for finite element partition of arbitrary shape: polygons or polyhedra F E n e x e A D A e C Figure: Illustration of a shape-regular polygonal element ABCDEFA. B

General WG Elements The general WG element has the following configuration for each v = {v 0,v b } v 0 P j (T) and v b P j 1 (e) or v b P j (e) V (T,r) = [P j 1 (T)] 2 The finite element space V h (j,j 1) consists of functions v = {v 0,v b } where v 0 is a polynomial of degree no more than j in T, and v b is a polynomial of degree no more than j 1 on T. The space V (T,r) used to define the discrete weak gradient operator w is given by polynomial space of order j 1 on T.

Depiction of the General WG Element P j 1 (e) P j 1 (e) P j (T) P j 1 (e) Figure: A triangular element with acute angles The space [P j 1 (T)] 2 is used for the computation of w,j.

III. Applications on WG FEM WG for Biharmonic Equation WG for Stokes Equations WG for Brinkman Equations

III-1. Biharmonic Equations (High order PDEs)

Biharmonic Equations Model problem: find an unknown function u satisfying 2 u = f in Ω, u = g on Ω, u = φ on Ω. n where is the Laplacian operator, Ω is a bounded polygonal or polyhedral domain in R d for d = 2,3 and n denotes the outward unit normal vector along Ω. We assume that f,g,φ are given, sufficiently smooth functions.

Variational Forms for Biharmonic Equations The weak form of the Biharmonic equations: seeking u H0 2(Ω) satisfying ( u, v) = (f,v), v H0(Ω), 2 Weak Galerkin finite element method: seeking u h V h satisfying ( w u h, w v) + s(u h, v) = (f, v), v V h. Key in WG: discrete weak Laplacian + stabilization in u to enforce weak continuities.

Weak Laplacian First we introduce weak functions on the element T. A weak function on T refers to a function v = {v 0,v b,v n } such that v 0 L 2 (T), v b L 2 ( T), and v n n L 2 ( T), where n is the outward normal direction of T on its boundary. The first component v 0 can be understood as the value of v in the interior of T, and the second and the third components v b and v n represent v and v on the boundary of T. Denote by V(T) the space of all weak functions on T; i.e., V(T) = {v = {v 0,v b,v n } : v 0 L 2 (T), v b L 2 ( T), v n n L 2 ( T)}.

Weak Laplacian Definition of Weak Laplacian The dual of L 2 (T) can be identified with itself by using the standard L 2 inner product as the action of linear functionals. With a similar interpretation, for any v V(T), the weak Laplacian of v = {v 0,v b,v n } is defined as a linear functional w v in the dual space of H 2 (T) whose action on each ϕ H 2 (T) is given by ( w v, ϕ) T = (v 0, ϕ) T + v n n, ϕ T v b, ϕ n T. where n is the outward normal direction to T.

Discrete Weak Laplacian Introduce a discrete weak Laplacian operator by approximating w in a polynomial subspace of the dual of H 2 (T). Definition of Discrete Weak Laplacian A discrete weak Laplacian operator, w,k, is defined as the unique polynomial w,k v P k (T) that satisfies the following equation ( w,k v, ϕ) T = (v 0, ϕ) T v b, ϕ n T + v n n, ϕ T for all ϕ P k (T). Shall drop the subscript k from w,k.

Weak Galerkin Algorithm Weak Galerkin Algorithm Find u h = {u 0,u b,u n n e } V h satisfying u b = Q b ζ and u n = Q b φ on Ω and the following equation: ( w u h, w v) h + s(u h,v) = (f,v 0 ), v = {v 0,v b,v n n e } V 0 h. Advantages: The triangle can be replaced by any polygon. Totally discontinuous functions of piecewise polynomials on general partitions makes lower order schemes can be used.

WG Elements I for Biharmonic The WG element for Biharmonic equation has the following configuration for each v = {v 0,v b,v n } v 0 P j+2 (T) and v b P j+2 (e) v n = v n n with v n P j+1 (e) w v P j (T). Mu-Wang-Ye, NM for PDEs, (2014) 59. Disadvantages: The freedom of bases brings the increasing of the number of unknowns. How to reduce the Dof?

Reduced Dof Schemes Existing Schemes for reducing Dof: Constructing C 0 -weak Galerkin (WG) method. Mu-Wang-Ye-Zhang, JSC, (2014) 59. Are there other schemes to reduce Dof? Adjusting stabilization to design an optimal combination of the polynomial spaces that minimizes the number of unknowns without compromising the rate of convergence for the corresponding WG method. Reducing the degree of freedom of WG-FEM to the degree on the edges(schur Complement skill). Zhang-Zhai, JSC, accepted.

Reduced Order WG Elements for Biharmonic The stabilization term for existing WG elements s(u h,v) = T T h h 1 T u 0 u n, v 0 v n T + T T h h 3 T u 0 u b,v 0 v b T. The stabilization term for the reduced order WG elements s(u h,v) = T T h h 1 T u 0 n e u n, v 0 n e v n T + T T h h 3 T Q bu 0 u b,q b v 0 v b T. Here Q h u = {Q 0 u,q b u,(q b ( u n e ))n e }.

Reduced order WG Elements for Biharmonic WG Elements I: {P k+2,p k+2,p k+1 } P j+2 P j+1 P j+2 P j+1 P j+2 (T) P j+2 P j+1 WG Elements II: {P k+2,p k+1,p k+1 } P j+1 P j+1 P j+1 P j+1 P j+2 (T) P j+1 P j+1

Schur Compliment of the WG formulation The WG method: find u h = {u 0,u b,u n n e } V h such that a(u h,v) = (f,v 0 ), v = {v 0,v b,v n n e } V h. For u h = {u 0,u b,u n n e } V h, solve for u 0 in term of u b,u n on T a(u h,v) = (f,v 0 ), v = {v 0,0,0} V h. Denote u 0 = D(u b,u n,f ). Find u b,u n such that a({d(u b,u n,f ),u b,u n n e,v) = 0, v = {0,v b,v n n e } V h. The system above: symmetric, positive definite, fewer unknowns.

Lowest Order WG Element for Biharmonic P 1 P 1 P 1 P 1 P 2 (T) P 1 P 1 The space P 0 (T) (i.e., piecewise constants) is used for the computation of w. Total dof: 6 + 3 (2 + 2) = 18 on each triangular element (compare with 21 for Argyris element P 5 on dof). Interior dof can be eliminated element wise. Thus, dof=12 for triangles.

Biharmonic Equations Example Consider the Biharmonic problem in the square domain Ω = (0,1) 2 : 2 u = f in Ω, u = 0 on Ω, u = 0 on Ω. n It has the analytic solution u(x) = x 2 (1 x) 2 y 2 (1 y) 2, and the right hand side function f is computed to match the exact solution. The mesh size is denoted by h = 1/n.

Biharmonic Equations Example 1

III-2. Stokes Equations (Saddle Points Problem)

WG method for the Stokes Equations As a model for the flow of an incompressible viscous fluid confined in Ω, we consider the following equations µ u + p = f, in Ω, (5) u = 0, in Ω, (6) u = g, on Ω, (7) for unknown velocity function u and pressure function p. Here µ is the fluid viscosity.

Weak Galerkin method for the Stokes equations Weak form of the Stokes equations: find (u,p) [H 1 0 (Ω)]d L 2 0 (Ω) that for all (v,q) [H1 0 (Ω)]d L 2 0 (Ω) ( u, v) ( v,p) = (f,v) ( u, q) = 0. Weak Galerkin method: find (u h,p h ) V h W h such that for all (v,q) V h W h ( w u h, w v) + s(u h,v) ( w v,p h ) = (f,v) ( w u h, q) = 0.

Weak Vector-Valued Functions Space of weak vector-valued functions: V(K) = {v = {v 0,v b } : v 0 [L 2 (K)] d, v b n L 2 ( K)}. A weak vector-valued function on the region K refers to a function v = {v 0,v b } such that v 0 [L 2 (K)] d and v b n L 2 ( K). The first component v 0 can be understood as the value of v in the interior of K, and the second component v b is the value of v on the boundary of K. v b may not be necessarily related to the trace of v 0 on K should a trace be defined.

Weak Divergence The dual of L 2 (K) can be identified with itself by using the standard L 2 inner product as the action of linear functionals. With a similar interpretation, for any v V(K), the weak divergence of v can be defined as a linear functional w v in the dual space of H 1 (K) whose action on each φ H 1 (K) is given by ( w v,φ) := v 0 φdk + (v b n)φds, K K where n is the outward normal direction to K. Weak divergence becomes to be strong divergence if weak functions are sufficiently smooth (e.g., as restriction of smooth functions).

Discrete Weak Divergence Weak divergence have to be approximated, which leads to discrete weak divergence w,r v given by ( w,r v)φdk = v 0 φdk + (v b n)φds, K for all φ W (K,r). Here K W (K,r) = P r (K) is the space of polynomials on K with degree r 0. W (K, r) should be the space for approximating the pressure component K

WG Elements for Stokes Problems [P j 1 ] d [P j 1 ] d [P j ] d P j 1 [P j 1 ] d The space P j (T) or P j 1 (T) can be used for the computation of w v. Lowest order element: pw linear for flux on each element, pw constant for pressure on element, pw constant for flux on edge. The triangle can be replaced by any polygon. Works for problems of any dimension.

Advantages of WG FEMs for Stokes The main strengthes of the WG method for Stokes are: the weak finite element space is easy to suit for any given stability requirement; the method allows the use of low order polynomials and hybrid meshes. It has competitive number of unknowns since lower degree of polynomials are used on element boundaries; the WG formulation is parameter free; the finite element partition can be of polytopal type. This property provides a convenient and useful flexibility in both numerical approximation and mesh generation;and the unknowns corresponding to the interior of each element can be eliminated from the system.

III-3. Brinkman Equations (Coupled Problems)

WG method for the Brinkman equations The Brinkman equations model fluid flow in complex porous media with a permeability coefficient highly varying so that the flow is dominated by Darcy in some regions and by Stokes in others. In a simple form, the Brinkman model seeks unknown functions u and p satisfying µ u + p + µκ 1 u = f, in Ω, (8) u = 0, in Ω, (9) u = g, on Ω, (10) where µ is the fluid viscosity and κ denotes the permeability tensor of the porous media which occupies a polygonal or polyhedral domain Ω R d (d = 2,3). u and p represent the velocity and the pressure of the fluid, and f is a momentum source term.

Challenge of algorithm design for the Brinkman equations The main challenge for solving Brinkman equations is in the construction of numerical schemes that are stable for both the Darcy flow and the Stokes flow. Stokes stable elements such as MINI element and Taylor-Hood element do not work well for the Darcy flow (small κ). Darcy stable elements such as RT elements and BDM elements do not work well for the Stokes flow (large κ).

WG method I for the Brinkman equations The weak formulation of the Brinkman equations: find (u,p) H 1 0 (Ω)d L 2 0 (Ω) such that for (v,q) H1 0 (Ω)d L 2 0 (Ω) (µ u, v) + (µκ 1 u,v) + ( v,p) = (f,v), ( u, q) = 0, The weak Galerkin finite element method for the Brinkman equations: find (u h,p h ) V h W h such that for (v,q) V h W h (µ w u h, w v) + (µκ 1 u h, v) + s(u h,v) ( w v, p h ) = (f, v), ( w u h, q) = (g, q), where s(v, w) = T T h h 1 T v 0 v b, w 0 w b T.

WG method II for the Brinkman equations The weak formulation of the Brinkman equations: find (u,p) H 1 0 (Ω)d L 2 0 (Ω) such that for (v,q) H1 0 (Ω)d L 2 0 (Ω) (µ u, v) + (µκ 1 u,v) + (v, p) = (f,v), (u, q) = 0, The weak Galerkin finite element method for the Brinkman equations: find (u h,p h ) V h W h such that for (v,q) V h W h (µ w u h, w v) + (µκ 1 u h, v) + s(u h,v) (v, w p h ) = (f, v), (u h, w q) = (g, q), where s(v, w) = T T h h 1 T v 0 v b, w 0 w b T.

Brinkman Equations Example 1 Figure: The values of κ 1. Analytic solution ( ) sin(2πx)cos(2πy) u = and p = x 2 y 2 1 cos(2πx) sin(2πy) 9. and κ 1 = a(sin(2πx) + 1.1), where a is a given constant.

Brinkman Equations Example 1

Brinkman Equations Example 1

Brinkman Equations Example 2 Example 2: (a) Profile of κ 1 for vuggy medium; (b) Pressure profile. Example 2: (c) First component of velocity u 1 ; (d) Second component of velocity u 2.

Brinkman Equations Example 3 Example 2: (a) Profile of κ 1 for fibrous stucture; (b) Pressure profile. Example 2: (c) First component of velocity u 1 ; (d) Second component of velocity u 2.

Brinkman Equations Example 4 Example 2: (a) Profile of κ 1 for open foam; (b) Pressure profile. Example 2: (c) First component of velocity u 1 ; (d) Second component of velocity u 2.

Challenge of algorithm design for the Brinkman equations L. Mu, J. Wang, and X. Ye A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J Comp Phys (2014) 273. Q. Zhai, R. Zhang and L. Mu A numerical study on the weak Galerkin method for the Brinkman equations. in Preparation. Our theory and numerical examples show that the weak Galerkin finite element solutions for the Brinkman equations converge uniformly independent of κ.

IV. Summary

Advantages of WG FEMs The weak Galerkin methodology provide a general framework for deriving new methods. the finite element partition can be of polytopal type; the weak finite element space is easy to construct with any approximation requirement; the weak finite element space is easy to suit for any given stability requirement; and the WG schemes can be hybridized so that some unknowns associated with the interior of each element can be locally eliminated, yielding a system of linear equations involving much less number of unknowns than what it appears.

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