HIGH-ORDER NUMERICAL METHODS FOR PRESSURE POISSON NAVIER-STOKES EQUATIONS

Transcription

1 HIGH-ORDER NUMERICAL METHODS FOR PRESSURE POISSON EQUATION REFORMULATIONS OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS DOCTOR OF PHILOSOPHY Dong Zhou August, 214 Benjamin Seibold, Advisory Chair, Mathematics Isaac Klapper, Mathematics Daniel Szyld, Mahematics Rodolfo Ruben Rosales, External Member, Massachusetts Institute of Technology, Mathematics

2 iii c by Dong Zhou August, 214 All Rights Reserved

3 iv ABSTRACT HIGH-ORDER NUMERICAL METHODS FOR PRESSURE POISSON EQUATION REFORMULATIONS OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS Dong Zhou DOCTOR OF PHILOSOPHY Temple University, August, 214 Professor Benjamin Seibold, Chair Projection methods for the incompressible Navier-Stokes equations (NSE) are e cient, but introduce numerical boundary layers and have limited temporal accuracy due to their fractional step nature. The Pressure Poisson Equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity, thus can be directly recovered by solving a Poisson equation, and no numerical boundary layers are generated; arbitrary order time-stepping schemes can be used to achieve high order accuracy in time. In this thesis, we focus on numerical approaches of the PPE reformulations, in particular, the Shiroko -Rosales (SR) PPE reformulation. Interestingly, the electric boundary conditions, i.e., the tangential and divergence boundary conditions, provided for the velocity in the SR PPE reformulation render classical nodal finite elements non-convergent. We propose two alternative methodologies, mixed finite element methods and meshfree finite di erences, and demonstrate that these approaches allow for arbitrary order of accuracy both in space and in time.

4 v ACKNOWLEDGEMENTS IwouldliketothankmyadviserProf. BenjaminSeiboldforhisguidance and assistance over the last four years. I am grateful to him for the opportunity to work on this interesting and challenging project, and for his support and trust throughout my dissertation research. Iwouldalsoliketothankallthecollaborators: Prof. RodolfoRuben Rosales (MIT), Dr. David Shiroko (McGill University) and Prof. Prince Chidyagwai (Loyola University Maryland), for their help during my dissertation research. It is a great pleasure working with all of them. IwouldalsoliketoexpressmygratitudetoProf. DanielSzyld,Prof. Issac Klapper and Prof. Rodolfo Ruben Rosales for serving on my dissertation committee, and for their comments and criticism during the writing of this thesis. Many thanks to Scott Ladenheim for proofreading my thesis draft, and to Shimao Fan and Stephen Shank for many inspiring conversations we have shared. I wish to acknowledge the National Science Foundation for the financial support through grant DMS Last but not least, I thank my parents and my friends for their support.

5 vi TABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENT LIST OF FIGURES LIST OF TABLES iv v ix xiii 1 Introduction The Navier-Stokes Equations Numerical Approaches Fully Implicit Approaches Projection Methods Pressure Poisson Equation Reformulations Pressure Poisson Reformulations for the Incompressible Navier- Stokes Equations The Pressure Poisson Equation Previously Proposed PPE Reformulations The Shiroko -Rosales PPE Reformulation PPE Reformulations as Extended Navier-Stokes Systems The Vector Poisson Equations with Electric Boundary Conditions The Vector Poisson Equations A Nodal Finite Element Approach Weak Formulations Numerical Implementation Failure of the Nodal Finite Element Approach Insu cient Regularity of Exact Solution Babuška Paradox Numerical Example

6 vii 4 Mixed Finite Element Methods for the Vector Poisson Problem and PPE Reformulation Mixed Finite Element Methods Function Spaces and Their Approximations Hilbert Spaces H(div) and H(curl) Approximations of H(div) and H(curl) Mixed Formulation for the Vector Poisson Problem Mixed Formulation for the SR PPE Reformulation Mixed Formulation for the Velocity System Weak Formulation of the Pressure Poisson Equation Semi-discretization of the Mixed Formulation Discretization of the Nonlinear Term Implicit-Explicit Time-stepping Schemes IMEX(1,1,1) IMEX(2,2,2) IMEX(3,4,3) IMEX(4,4,3) Time Discretization of the Mixed Formulation Streamline Upwind/Petrov-Galerkin-Type Stabilization Streamline Upwind/Petrov-Galerkin Formulation SUPG for Linear Advection-Di usion Equation SUPG for Linear Advection Equation SUPG Modification of the Mixed Formulation Vector Linear Advection-di usion Equation Mixed Formulation of the SR PPE Reformulation The Stabilization Parameter In the Absence of Stabilization Influence of the Stabilization Parameter Numerical Results A Manufactured Solution Flow on a Square Domain Flow on Irregular Domains Influence of the Parameter SUPG Stabilization Lid-driven Cavity Influence of the Parameter Backward-facing Step Flow Preliminary Results on Flow Around a Cylinder

7 viii 7 Meshfree Finite Di erences for the Vector Poisson Problem and PPE Reformulations Meshfree Finite Di erence Method Consistent Derivative Approximations Least Squares Approaches Point Cloud Generation Meshfree Finite Di erences for the Vector Poisson Equation Meshfree Finite Di erences for the Vector Heat Equation Implicit Approaches Explicit Approaches Meshfree Finite Di erences for the SR PPE Reformulation The Velocity System The Pressure Poisson Equation Approximate Laplacian at Domain Boundary Meshfree Finite Di erences for the JL PPE Reformulation Numerical Results The Vector Poisson Equation The Vector Heat Equation The SR PPE Reformulation The JL PPE reformulations Conclusions and Outlook Conclusions A Mixed Finite Element Approach A Meshfree Finite Di erence Approach Outlook REFERENCES 138

8 ix LIST OF FIGURES 3.1 Domain with re-entrant corner at the origin The true vector field (red arrows) and numerical solutions (black arrows) obtained via nodal finite element method. The solutions obtained with two di erent mesh resolutions are almost identical, and the error convergence plot also shows that the FEM solution di ers from the true solution Basis functions of the lowest order Raviart-Thomas elements (RT )definedonareferencetriangle Basis functions of lowest order Nédélec elements (NED 1 )defined on a reference triangle Global shape functions for lowest order Raviart-Thomas (left) and Nédélec (right) finite element spaces in two dimensions. The normal component of the RT global shape function is continuous across the edge, whereas the shape function for NED space is continuous in tangential direction and the normal component can have jumps Uniform triangular meshes for computing the eigenvalues of the update matrix for the vector linear advection-di usion problem Eigenvalues of the update matrix for various values of with a2 nd order finite element discretization and a mesh with 16 elements. Without stabilization, the eigenvalues move to the right half-plane as gets smaller (advection dominates) and the method becomes unstable Eigenvalues of the update matrix for various values of with a3 nd order finite element discretization and a mesh with 16 elements. Without stabilization, the eigenvalues move to the right half-plane as gets smaller (advection dominates) and the method becomes unstable

9 x 5.4 Motion of the eigenvalues of the update matrix with respect to the change of C. IMEX(1,1,1) with 2 nd order finite element discretization, C 2 [, 1.1] and =.1. Bigger values of C correspond to darker shades of red Region of (,C) such that all the eigenvalues of the update matrix are in the left half-plane. Black curves outline the boundary of the region for the 3 rd order spatial discretization on two meshes; blue curves outline the boundary of the region for the 2 nd order spatial discretization Error convergence for manufactured solution Error convergence for manufactured solution with the choice of finite element spaces P r RT r 1 P r 1, r =2, Error convergence for manufactured solution with the choice of finite element spaces P r RT r 1 P r+1, r =2, Divergence at final time T = using a 3 rd -order scheme on a regular mesh with 124 triangular elements Sample meshes for domain 1 (left) and 2 (right) Error convergence result on the unit square domain (top left), 1 (top right) and 2 (bottom) Error convergence for manufactured solution using 3 rd -order approach with = and = Comparison of di erent values of stabilization parameter for 2 nd and 3 rd -order schemes for = Streamline plot of lid-driven cavity flow with Re =1atsteady state (left) and velocity through the centerline compared with reference data (right). Our numerical results are shown in solid and dashed lines, and the reference data is shown in red and blue circles Streamline plot of lid-driven cavity flow with Re =4atsteady state (left) and velocity through the centerline compared with reference data (right). Our numerical results are shown in solid and dashed lines, and the reference data is shown in red and blue circles Streamline plot of lid-driven cavity flow with Re = 1at steady state (left) and velocity through the centerline compared with reference data (right). Our numerical results are shown in solid and dashed lines, and the reference data is shown in red and blue circles (a) Convergence of kr uk L 1 ( ) and (b) Flow through the boundary, for di erent values of

10 xi 6.13 Plot of r u in log-scale for lid-driven cavity test with Re = 1 and = Velocity field near the top right corner. Velocity field (left) and normalized velocity field (right). From top to bottom correspond to =.1, Velocity field near the top right corner. Velocity field (left) and normalized velocity field (right). From top to bottom correspond to =1, Schematic view of the backward-facing step test Sample mesh for computing backward-facing step test Streamline plot for Re =1(top),Re =2(middle),Re = 4 (bottom) Mesh used for computing flow around a cylinder test Plot of the vorticity at times t =2, 4, 5, 6, 7, Computed drag (top left), lift (top right) and pressure di erence (bottom) as a function of time, for apple t apple Point cloud for the computational domain with 1 points. The boundary points are shown in red and the interior points in blue Mesh size for a point cloud defined by hexagonal tiling. The gray small hexagon gives the point area A for each point Plot of the approximation error in u for two approaches for evaluating u at the boundary. Black dots: use finite di erences directly at boundary points. Red dots: extrapolate function u from interior to the boundary via the MLS Error convergence for 1 st (top left), 2 nd (top right), and 3 rd (bottom)-order meshfree finite di erences approximations. The errors are measured in the maximum norm. The results show that n th -order meshfree finite di erences stencils result in n th - order convergence rates for the solution of the VPE, and its derivatives. The convergence occurs all the way to the boundary, even for the derivatives Error convergence for the vector heat equation, using a spatially second order meshfree finite di erence discretization. Top left panel: using forward Euler time stepping with k =.2h 2, confirming the expected O(h 2 )convergenceorder. Toprightpanel: using backward Euler time stepping with k = 1h, yielding the expected O(h) convergenceduetotemporalerrors. Bottom panel: using IMEX(2,2,2) scheme with k = h, confirming the expected O(h 2 )convergenceorder

11 xii 7.6 Error convergence of the velocity field, the pressure, and their derivatives, for the PPE reformulation of the NSE. A spatially second order meshfree finite di erences discretization is used. Left panel: forward Euler time stepping (with t =.2h 2 ). Right panel: IMEX(2,2,2) time stepping (with t =.2h). In both cases, the expected O(h 2 )convergenceisconfirmed Lid-driven cavity test with Re=1 using a second-order meshfree scheme with 4 points. Left: normalized velocity field at time T =2. Right: plotoftheflownormaltothetwo centerlines of the cavity compared with the reference data [42] Error convergence for the JL PPE reformulation with a 2 nd - order spatial discretization and a 1 st -order IMEX scheme

12 xiii LIST OF TABLES 6.1 Computed detachment and reattachment point for Re =1, 2 and 4 compared with reference values (in parentheses) from [35]

13 1 CHAPTER 1 Introduction In this thesis, we propose two numerical methods for solving the incompressible Navier-Stokes equations and some related equations that are motivated by the pressure Poisson equation reformulation of the Navier-Stokes equations. The numerical methods are based on a particular pressure Poisson equation reformulation of the incompressible Navier-Stokes equations by Shiroko and Rosales [98]. 1.1 The Navier-Stokes Equations The Navier-Stokes equations (NSE) have long been the center of computational fluid dynamics (CFD) due to their practical importance in modeling fluid motion. The equations were named after Claude-Louis Navier [84] and George Gabriel Stokes [99], and have been used to describe viscous fluid. In continuum mechanics, the motion of a fluid is often characterized by its velocity field. By employing the conservation of mass and balance of momentum, asystemofdi erential equations can be derived. Furthermore, under the assumptions that the stress depends on the velocity deformation tensor linearly and the fluid is incompressible, which are satisfied for instance in the case of

14 2 water, one obtains the incompressible Navier-Stokes equations u t + (u r)u = µ u rp + f, r u =, where denotes the density which is a constant in the case of a incompressible fluid, µ is the dynamic viscosity, p is the pressure and f represents the body forces such as gravity. Let p = p, = µ (kinematic viscosity), f = 1 f. Then, dropping the primes, we obtain a widely presented form of the incompressible NSE u t +(u r)u = u rp + f, (1.1) r u =. (1.2) Although the incompressible NSE are of great practical interest, the problem of proving existence and smoothness of the solution in 3D still remains open, and is one of the Clay Institute Millennium prize problems [36]. Nevertheless, the NSE (1.1) (1.2) are solved numerically in practice, thus devising accurate and e cient numerical solvers for such problems has attracted a lot of attention in the computational science community. 1.2 Numerical Approaches Developing numerical methods for solving the NSE (1.1) and (1.2) has its own challenges and di culties. A fundamental question is how to implement the incompressibility constraint. Or equivalently, given a velocity field at any instance in time, how to recover the pressure. Since the pressure is introduced into the equations as a Lagrange multiplier for the incompressibility constraint, there is no time evolution for the pressure, hence there is no single canonical way to numerically advance the NSE forward in time. Here we give a brief overview of some common and popular numerical solution strategies for the incompressible NSE including fully implicit formulations and projection methods. Then we introduce a less well known class of methods, which have the

15 3 potential for devising high-order and e Poisson equation (PPE) reformulations. cient numerical schemes, the pressure Fully Implicit Approaches One class of numerical approaches to advance the NSE is based on approximating the time derivative in the momentum equation, and solving for u and p in a fully coupled fashion [73]. For example, discretizing in time via backward Euler time-stepping scheme, one can obtain the velocity and the pressure (u n+1,p n+1 )attimet n+1 = t n + k by solving 1 k (un+1 u n ) u n+1 +(u n+1 r)u n+1 + rp n+1 = f(t n+1 ), r u n+1 = given the velocity at time t n, u n. Using some spatial discretization, one obtains anonlinearsystemwithsaddlepointstructure A(u n+1 ) B T! B O u n+1 p n+1! = F!, (1.3) where O denotes the zero matrix. Standard Newton or Picard iteration methods can be used for solving such nonlinear problems [57, 65]. Although the fully implicit approach is accurate and treats every situation correctly, it is computationally costly since a nonlinear system that possesses a saddle point structure must be solved at each time step. While the system (1.3) can be linearized by using the velocity from the previous time step, the resulting discrete system may sometimes be singular. Extra care needs to be taken in the spatial discretization to ensure the invertibility of the matrix. A straightforward central di erencing finite di erence approach on a regular grid leads to the checkerboard instability [88]. The remedy for this in the finite di erence context is to use the staggered grid introduced by Harlow and Welch [48] in the Marker-and-Cell (MAC) scheme. By employing a staggered grid, the velocity components and the pressure are

16 4 solved at di erent points in the grid. In the finite element context, an analogy of this leads to mixed finite element approaches in which di erent finite elements are used for the velocity and the pressure. A key to ensure that the discrete system is non-singular is the discrete inf-sup condition due to Ladyzhenskaya, Babuška and Brezzi. The fully coupled system with a saddle point structure is also challenging for developing linear solvers. A large amount of work has been devoted to develop e cient solution methods for such linear systems [17] Projection Methods Projection methods are very popular for time-dependent incompressible viscous flows in applications. The methods decouple the velocity and the pressure in each time step, which makes them very e problems. cient for large-scale The original scheme was proposed by Chorin [27] and Temann [1] in the late 196s. It was formulated in a time splitting form: one time step of using forward Euler time-stepping consists of two sub-steps, assuming homogeneous Dirichlet boundary condition for the velocity: 1. Advance the velocity using the momentum equation (1.1) without the pressure 1 k (u u n )= u n +(u n r)u n + f n. 2. Correct the intermediate solution u by a pressure 1 k (un+1 u )= rp n+1, such that the new velocity u n+1 satisfies r u n+1 =intheinteriorof the domain and n u n+1 = at the boundary. This leads to a Poisson equation for the pressure p n+1 with a Neumann boundary condition 8 >< p n+1 = 1 k r u in, = on@.

17 5 In step 2, the velocity is projected onto the space of divergence-free fields via a Poisson problem. Unfortunately, the artificial Neumann boundary condition for the pressure introduces numerical boundary layers which cause degradation in error convergence [46]. The scheme described above also inherits a splitting error of order O(k), which limits the overall accuracy of the scheme even if higher-order time-stepping schemes are used in step 1. An extensive overview of projection methods is given by Guermond, Minev and Shen in [46], where projection methods and various improvements, as well as their theoretical and numerical convergence results are discussed. Alotofe orts have been made to improve projection methods over the past few decades. Second order projection methods have been developed [15, 67, 68]. However, the issue of boundary layers still remains [46] Pressure Poisson Equation Reformulations In the pressure Poisson equation (PPE) reformulation of the Navier-Stokes equations [49, 51, 52, 63, 71, 92, 93, 98], a Poisson equation for the pressure is derived to replace the incompressibility constraint r u = byaglobal pressure function p = P (u). The global pressure function is designed so that the solutions of the PPE reformulation are identical to the solutions of the original NSE. To obtain such P (u), a boundary condition for the pressure must be selected. The choice of boundary conditions for the pressure has been discussed extensively, both in the PPE reformulations and in the projection methods context, see [43, 67, 93]. Henshaw [49] pointed out that r u = is a natural condition to add in order to obtain an equivalent reformulation to the NSE. In a sequence of work by Henshaw et al. [49, 5, 51, 52], the recovery of the pressure through the solution of a pressure Poisson problem was done in a discrete setting using finite di erences, and the divergencefree boundary condition is applied, together with other numerical boundary conditions. A continuous formulation was later introduced in [64] by Johnston and Liu. Recently, a di erent PPE reformulation that imposes r u =as

18 6 boundary condition together with tangential boundary condition was proposed by Shiroko and Rosales [98]. The advantages of PPE reformulations compared to the standard form of NSE are: 1) the pressure is no longer implicitly coupled to the velocity, and can be directly recovered by solving a Poisson equation; 2) no numerical boundary layers are generated as in projection methods [98]. As a consequence of 1), numerical methods based on PPE reformulations decouple the pressure solve and the velocity solve, and are structurally easy to extend to high order accuracy in time. A di culty of PPE reformulations is that the Poisson equation for the pressure can involve complicated expressions, whose interaction with the velocity field equation is not always easy to understand and to analyze. Other important properties of PPE reformulations are that, the divergence field is not exactly zero if solved numerically, and unlike the original NSE, the PPE reformulations are also defined even if the initial conditions are not incompressible. We introduce and compare some PPE reformulations in Chapter 2, with particular interest in the Shiroko -Rosales (SR) PPE reformulation. The boundary condition for the velocity (electric boundary conditions) in the SR PPE reformulation poses di culties in devising numerical methods. In Chapter 3, we demonstrate via a model problem, the vector Poisson equation with electric boundary conditions, that a standard nodal-based finite element approach fails and a Babuška paradox can occur. Thus in Chapter 4, we propose a mixed finite element approach that treats this problem correctly and achieves high order accuracy both in space and in time. In Chapter 5, we focus on the issue that the mixed finite element method is unstable when the Reynolds number is large, i.e., the flow becomes advection-dominated. We propose a modification motivated by the streamline upwind/petrov-galerkin method, but in the context of mixed finite element methods. In Chapter 6, we perform various numerical experiments testing the convergence of the scheme, conduct benchmark tests and compare results with reference data.

19 7 Lastly, we present an alternative method for the SR PPE reformulation. This approach is based on meshfree finite di erences. We demonstrate that this approach is conceptually simpler than the mixed finite methods, but the resulting scheme also achieves high order accuracy both in space and in time and can be easily extended to other PPE reformulations.

20 8 CHAPTER 2 Pressure Poisson Reformulations for the Incompressible Navier-Stokes Equations In this chapter, we introduce the principle of pressure Poisson equation (PPE) reformulations of the incompressible Navier-Stokes equations and provide an overview of di erent PPE reformulations. We are particularly interested in the PPE reformulation proposed by Shiroko and Rosales, on which our numerical methods are based. 2.1 The Pressure Poisson Equation We consider the time-dependent incompressible Navier-Stokes equations (NSE) in a connected domain 2 R d, where d =2or3,withpiece-wise

21 9 smooth u t +(u r)u = u rp + f in (,T] (2.1a) r u = in (,T] (2.1b) u(x,t)=g(x,t) [,T] (2.1c) u(x, ) = u (x) in (2.1d) with the compatibility conditions Continuity between the initial condition and the boundary condition: u (x) =g(x, ) (2.2) Incompressibility for the initial condition: r u =in. (2.3) Zero net flux through the boundary: Z n g ds =. Here is the kinematic viscosity. Equation (2.1a) follows from the conservation of momentum, and (2.1b) is conservation of mass (incompressibility condition for constant density). A general way of deriving a PPE reformulation of the NSE is: take divergence of the momentum equation (2.1a) and apply the incompressibility condition (2.1b) to eliminate the viscous term and the time derivative term. This yields the pressure Poisson equation p = r (f (u r)u). (2.5) So the question remains what boundary conditions to prescribe for the pressure Poisson equation such that the pressure can be determined for given velocity, while the incompressibility condition is preserved? This has led to

22 1 much discussion [43, 49, 63, 64, 67, 98]. Candidate boundary conditions for the pressure can be derived from the momentum equation. The pressure can therefore be formulated as a global function of the velocity, p = P (u). In addition, the boundary condition for the momentum equation u t +(u r)u = u rp (u)+f (2.6) must be selected so that the solutions of the PPE system are equivalent to the ones of the standard formulation of the NSE. Equation (2.6) can then be viewed as an evolution equation for the velocity, which could be solved using high-order time integration methods. In the following sections, we discuss di erent PPE reformulations in more detail. 2.2 Previously Proposed PPE Reformulations One of the early forms of PPE reformulations was due to Henshaw et al. [49, 5, 51, 52] in which the divergence boundary condition is prescribed in addition to the Dirichlet boundary condition. This leads to a velocity-pressure formulation of the form u t +(u r)u = u rp (u)+f in (,T] (2.7a) p = r (f (u r)u) in (2.7b) u = g [,T] (2.7c) r u = on@ [,T] (2.7d) The extra boundary condition (2.7d) is added to make the problem well-posed [49]. In the finite di erences setting, a numerical scheme is proposed by adding numerical boundary conditions u t +(u r)u = u rp (u)+f (2.8a) p = r (f (u r)u) r u = on@.

23 11 These boundary conditions are required in order to determine the velocity and the pressure at two lines of ghost points in their high-order numerical approximations. In their actual numerical scheme, the numerical boundary condition (2.8a) is split into normal and tangential components, and a discrete version of the pressure Poisson equation with pure Neumann boundary condition (normal component of (2.8a)) is solved, the remaining boundary conditions are used to determine the velocity. The discretization is done in such a way that the velocity and the pressure are decoupled, and an explicit time-stepping scheme can then be used to advance the velocity forward in time. Remark 2.1. Although the pressure boundary condition (i.e., normal component of (2.8a)) is introduced as a numerical boundary condition for the purpose of discretization, the approach decouples the update of the velocity and the pressure in a way that is di erent from the projection methods. For solutions that are smooth enough, taking the divergence of (2.7a) and using (2.7b) reveals that the divergence = r u satisfies the heat equation t = in the interior of the domain. Together with the boundary condition (2.7d), the divergence is guaranteed to be zero for all time provided that (t =)=. Remark 2.2. Further modifications were made by Henshaw and Petersson [52] where a divergence damping term (x)r u, is added to the pressure equation (2.7b) to obtain a new pressure Poisson equation, p = r (f (u r)u)+ (x)r u. The result is that the divergence satisfies the PDE t =.

24 12 Of course, in the continuous case, adding the damping term has no e ect since r u =, however in the discrete case where the divergence is not exactly zero, the damping term adds an exponential decay to the divergence and e ectively keeps the discrete divergence small. Henshaw and Petersson [52] also discussed two di erent choices of Neumann = n (f u t (u r)u + = n (f u t (u r)u r r u), (2.1) for the discrete pressure system. A parabolic scaling of time step k / h2 is required for (2.9) even if the viscous term is treated implicitly. A detailed stability analysis for both pressure boundary conditions is given by Petersson in [9]. The PPE reformulation we shall discuss next, proposed by Johnston and Liu [64], is a continuous analogue to the choice of pressure boundary condition (2.1) in the scheme by Henshaw and Petersson. The PPE reformulation by Johnston and Liu [64] (we will refer to it as the JL PPE reformulation) is of the form u t +(u r)u = u rp (u)+f in (,T], (2.11a) u = g [,T], (2.11b) where P (u) isasolutionoftheassociatedpressurepoissonproblem Let p = r (f (u r)u) = n (f g t r r u (u r)u) (2.12b) = r u, then applying divergence to (2.11a) and using (2.12a) yields the same heat equation for in the interior of the domain as in Remark 2.1. Dotting (2.11a) at the with n and using (2.12b) leads to n ( u)+n ( r r u) = on@. (2.13)

25 13 The boundary condition for follows from applying the vector identity u = r(r u) r r u to (2.13). The divergence satisfies the heat equation with homogeneous Neumann boundary condition 8 >< t = in (2.14) = on@ [,T]. If =initially,i.e.,theinitialvelocityfieldisincompressible,thenequation (2.14) ensures that =(incompressibility)atalltime. Variations of the JL PPE reformulation and numerical implementations have been discussed in [79, 8, 81, 82]. In [81], Liu, Liu and Pego formulate and analyze a finite element approach of the JL PPE reformulation using C 1 elements. A finite element version of the projection step is incorporated into the scheme to suppress the divergence in some practical test such as backward-facing step flow. The JL PPE reformulation has been extended to more complicated boundary conditions for the NSE in [79], where pseudotraction boundary conditions n ru pn = g and the boundary condition that prescribes force acting on part of the boundary (ru + ru T ) n pn = g are considered. Remark 2.3. The projection step in [81] is referred to as divergence suppression, and interestingly, the resulting numerical scheme is a finite element analogue to Henshaw and Petersson s approach [52] with the damping term r u, where / 1 k and k is the time step length.

26 2.3 The Shiroko -Rosales PPE Reformulation We are interested in a particular PPE reformulation of the NSE proposed in [98], namely the Shiroko -Rosales PPE (SR PPE) reformulation. Its fundamental di erence from previously proposed PPE reformulations is that the velocity field satisfies electric boundary conditions, i.e., incompressibility and the tangential flow are prescribed at the boundary. In turn, the normal velocity is enforced via a relaxation term in the pressure equation. The SR PPE reformulation consists of the momentum equation u t +(u r)u = u rp (u)+f in (,T], (2.15a) n u = n g [,T], (2.15b) r u = on@ [,T], (2.15c) where P (u) isasolutionoftheassociatedpressurepoissonequation p = r (f (u r)u) = n (f g t + u (u r)u)+ n (u g) (2.16b) Here the term n (u g) isarelaxationtermaddedtofurtherenforcethe normal velocity at the The normal velocity at the boundary is enforced implicitly through an ordinary di erential equation 8 < n (u t g t )= n (u g) (,T] : n (u = att =, 14 (2.17) which is obtained by dotting (2.15a) with n, evaluating at the boundary and using the pressure boundary condition (2.16b). Equation (2.17) has the exact solution n (u the parameter g) =andinthepresenceofnumericalapproximationerrors, (> ) adds an exponential decay to the error in the normal velocity. Notice that (in the absence of numerical approximation errors) equation (2.17) would still return an exact solution n (u g) =withoutthe

27 15 -term. However, in practice, numerical errors result in a drift of the normal velocity [98], which can be stabilized by adding the -term. For smooth enough (up to the boundary) solutions (u,p), it is shown in [98] that the NSE (2.1) are equivalent to the SR PPE reformulation (2.15) (2.16). Remark 2.4. Similar to the PPE reformulation (2.11), (2.12), the divergence = r u for the above SR PPE reformulation also satisfies the heat equation in the interior of the domain, but with a homogeneous Dirichlet boundary condition which is due to the direct enforcement of the divergence boundary condition (2.15c) 8 < : t = in = (2.18) Again, the equation (2.18) ensures the incompressibility at all time given = initially. If, due to numerical approximation errors, the numerical solution starts to drift away from the r u =manifold, equation (2.18) also ensures that it is pulled back towards incompressibility. This property indicates that in PPE reformulations, there is no need to impose a discrete incompressibility principle. Remark 2.5. (Solvability condition for the pressure Poisson equation) The pressure Poisson problem (2.16) has a solution only if the solvability condition is satisfied, i.e., if Z (r u)+ r u dv n g t + n g ds =. The boundary integral vanishes due to the zero net flux condition (2.4), and the volume integral is zero only if r u =. In the presence of numerical approximation errors, or because a problem with r u 6=is considered, the solvability condition may be violated. However, whenever this occurs, one can solve an augmented system that projects the right hand side of (2.16) in a way that the solvability condition is satisfied; this approach will be discussed later in

28 16 Our goal is to develop numerical methods for the SR PPE reformulation on arbitrary domains. The challenge lies in how to implement the boundary conditions properly. The boundary conditions for the velocity in the SR PPE reformulation consist of enforcing the tangential component of the velocity, together with the divergence free condition. Due to their occurrence in electrostatics, they are often called electric boundary conditions, a terminology we adopt here. 2.4 PPE Reformulations as Extended Navier- Stokes Systems Although PPE reformulations originally arose in the numerical context, as a means to solve the incompressible Navier-Stokes equations, they can be viewed as extended Navier-Stokes systems in which the problem is still wellposed even for velocity fields that are not divergence-free. The term extended Navier-Stokes system was introduced in [81] where a finite element approach based JL PPE reformulation was proposed. Later, both the JL and the SR PPE reformulations are discussed and analyzed as extended Navier-Stokes systems from a theoretical point of view in [58, 59]. Since the solvability of the pressure system of the SR PPE reformulation (2.16) relies on the divergence-free condition of the velocity, and neither the initial data nor the solution are required to be divergence-free for the extended system, the SR PPE reformulation takes on a slightly di erent form when discussed in extend Navier-Stokes system setting: u t +(u r)u = u rp (u)+f in (,T], (2.19a) n u = n g [,T], (2.19b) r u = on@ [,T], (2.19c)

29 17 where P (u) isasolutionof p = r (f (u r)u) = n (f g t + u (u r)u)+ n (u g)+c (2.2b) The extra term C = C(t) isdefinedas C = 1 Z n ( u + = 1 Z + where = r u. It is added to satisfy the solvability condition for the pressure. It is pointed out in [98] that this modification not only allows divergence being nonzero, but relaxes the normal boundary condition to satisfy the ODE It is clear that if n (u t g t )= n (u g)+c = r u is initially zero, it will remain zero for all time due to (2.18), and the system is reduced to the SR PPE reformulation (2.15) and (2.16) since C =. Remark 2.6. Notice that there are other possible forms of the extended Navier- Stokes system. For example, the nonlinear term and the viscous term can be expressed in divergence form. The nonlinear term in divergence form is r (u u), (2.21) where denote the tensor product or outer product of vectors. Due to the vector identity r (u u) =(u r)u +(r u)u, the expression (2.21) is equivalent to the nonlinear term (u r)u provided r u =. The viscous term is derived from 2 r (u), (2.22)

30 18 where (u) = 1(ru + 2 rut ) is the strain rate tensor. When the flow is incompressible, i.e., r u =, we have 2 r (u) = ( u + r(r u)) = u. Using (2.21) and (2.22) in the above extended system has the advantage that the system is in conservation form. Errors caused by r u 6= will not produce extra momentum. However, the modified system may not remain to be an extended Navier-Stokes system, and modifications of the boundary conditions may be required.

31 19 CHAPTER 3 The Vector Poisson Equations with Electric Boundary Conditions In this chapter, we consider the vector Poisson equation with electric boundary conditions, u = f in, r u = on@, n u = n g where r f =,asamodelproblemanddiscussnumericalmethodsforsolving this problem. This is the first step towards developing numerical methods for the SR PPE reformulation introduced in the previous chapter. The vector Poisson equation is the steady state limit of the vector heat equation u t u = f in (,T], r u = on@ (,T], n u = n g (,T], u = u in {t =},

32 2 which is a simple analog of the momentum equation in the SR PPE reformulation. The electric boundary conditions pose challenges in devising numerical methods. We will show that classical nodal finite element methods in this case converge to the wrong solution and that a Babu ska paradox can occur. 3.1 The Vector Poisson Equations The vector Poisson equation (VPE) arises, for instance, in electrostatics problems. The electric field satisfies r E =, where = (x) isthecharge density. Using the condition that r E =impliesthat E = r in the VPE. Moreover, if the boundaries of the domain are perfect conductors, then the vector field is perpendicular to the boundary, i.e., n E =,wherenis the normal vector to the outer surface of the domain. Another example comes from the magnetic potential, which satisfies the VPE, A = J, where J = J(x) is the electric current density. The Coulomb gauge yields, r A =,andthe boundary condition n A = representsazeromagneticfield(see[32]for more details). Motivated by the structure of these examples, we consider here the VPE u = f in, (3.3a) r u = on, (3.3b) n u = n g (3.3c) The divergence constraint often cause challenges in numerical approaches, therefore it is desirable to remove the divergence condition that is imposed in the whole domain. The following outlines how this can be achieved. Let be a bounded, simply connected domain with Lipschitz and n be the outward unit normal vector along the boundary. The vector

33 21 Poisson equation (VPE) with electric boundary conditions is of the form u = f in, (3.4a) r u = on@, (3.4b) n u = n g (3.4c) where the source function is incompressible, i.e., r f =. Note that in contrast to problem (3.3), problem (3.4) possesses no divergence-free condition in the interior of the domain. Instead, r u =isspecifiedasanadditionalboundary condition. Clearly, any solution of (3.3) is also a solution of (3.4). Moreover, we have the following result. Lemma 3.1. If the solution to (3.4) is in (H 2 ( )) d, then it is also a solution to (3.3). Proof. Define = r u. Then 2 H 1 ( ) isa(weak)solutionoftheproblem 8 < = in (3.5) : = on@, which has the unique solution. Hence r u =in. 3.2 A Nodal Finite Element Approach Among possible approaches for numerically approximating the problem (3.4) on an irregular domain, a natural first choice would be standard nodalbased finite elements. In this section, we derive a possible weak formulation of the VPE with electric boundary conditions and discuss the numerical method when using nodal-based finite elements Weak Formulations We first introduce a weak formulation of the VPE (3.4). It is natural to work with the a ne Hilbert space of vector-valued H 1 functions that satisfy

34 22 the tangential boundary condition (3.4c), Hgt( ) 1 d = {u 2 H 1 ( ) d : n (u =}. Moreover, let Ht( ) 1 d denote the associated homogeneous (i.e., g = ) Hilbert space. Using the identity, u = r (r u) r(r u), and following the standard procedure, we multiply (3.4a) by a test function v 2 H t ( ) d and integrate by parts, which gives hf, vi = h u, vi = hr r u, vi hr(r u), vi Z = hr u, r vi v (r u) n ds + hr u, r vi (r u)(v n) = hr u, r vi + hr u, r vi Z Z (r u) (n v) ds (r u)(v Since u and v satisfy r u =andn v =on@, we obtain the bilinear form Z a(u, v) = (r u) (r v)+(r u)(r v) dv. (3.6) The weak formulation of (3.4) reads as: given f 2 L 2 ( ) d with r f =,find u 2 H gt ( ) d such that for each v 2 H t ( ) d a(u, v) =hf, vi. (VP1) Note that the condition, (r =, arises as a natural boundary condition: assume that u is a weak solution to (VP1) andissu satisfies hf, vi = a(u, v) =h ciently smooth, then u Z Z u, vi+ (r u) (n v) ds + (r u)(v

35 23 for all v 2 H 1 t( ) d. The first integral vanishes due to n =. Hence, we obtain u = f in, andr u =on@ Numerical Implementation Based upon the weak formulation (VP1), the divergence boundary condition emerges as a natural boundary condition, and we need only consider the other part of the electric boundary conditions, namely the tangential boundary condition. In first order nodal-based finite elements, the degrees of freedom are associated to the values at the nodes of the computational mesh. higher-order elements, there are degrees of freedom associated to the point values on the edges. At the boundary vertices, the notion of a normal vector is not clearly defined. Here, we adopt the approach of implementing tangential boundary conditions introduced by Engelman and Sani in [33]. There they defined the normal vectors as suitable averages of normal vectors at the edges connecting to the boundary node. To implement a solution method for the weak formulation (VP1) in 2D, we consider vector-valued basis functions v =! and w =!, For where and are the standard nodal-based finite elements, or Lagrange finite elements [28]. Let I and J denote the set of all interior nodes and the set of all boundary nodes, respectively. The numerical solution is represented by a linear combination of basis functions u = NX u i v i + v i w i with n 2 ku k n 1 kv k = n 2 kg 2 n 1 kg 1 = g k 8k 2 J, (3.7) i=1 where N is the total number of nodes (interior and boundary nodes),! n k = n1 k n 2 k

36 24 is the normal vector at boundary node k and g k is the corresponding tangential boundary condition. When formulating the linear system, we first make no distinction between interior nodes and boundary nodes, therefore the test function space is span{v j, w j j 2 I} for interior nodes, span{v k, w k k 2 J} for boundary nodes. The unknowns are arranged into a vector of length 2N 1 U I V I U J V J where U I, V I are the vectors of velocity components at the interior nodes, and U J, V J are the vectors of velocity components at the boundary nodes. Testing the approximate solution against all basis functions, we obtain a 2N 2N large, sparse linear system A vi v I A wi v I A vj v I A wj v I U I F vi A vi w I A wi w I A vj w I A wj w I V I F wi =. A vi v J A wi v J A vj v J A wj C B v J U J C B F vj C A A vi w J A wi w J A vj w J A wj w J V J F wj where A vi w J denotes the matrix with entry A ij = a(v i, w j )withi2i and j 2 J,andF vi is a vector with each component being hf, v i i, i 2 I. The tangential boundary condition is enforced by modifying the last two block-rows of the matrix. This procedure is done by replacing the two equations at every boundary node k 2 J a(u h, v k )=hf, v k i a(u h, w k )=hf, w k i, C A

37 25 by the following two equations n 2 ku k n 1 kv k = g k (tangential b.c. at boundary node k) (3.8) a(u h,n 1 kv k + n 2 kw k )=hf,n 1 kv k + n 2 kw k i. (3.9) Equation (3.9) can be interpreted as preserving the normal component at the boundary n ( u) =n f in a discrete sense. The modified linear system, still of size 2N 2N, becomes 1 1 A vi v I A wi v I A vj v I A wj v I U I A vi w I A wi w I A vj w I A wj w I V I O O diag(nj 2 ) diag(n J 1 ) = C B U J C B A vi N J A wi N J A vj N J A wj N J V J F vi F wi G J F NJ 1 C A. The vectors NJ 1 and N J 2 consist of the first and the second component of the normal vectors at all boundary nodes respectively. The matrix A vi N J has entries a(v i,n 1 k v k + n 2 k w k)foralli2i, k 2 J. G J is the vector of tangential boundary condition at all boundary nodes and F NJ is the vector with each component being hf,n 1 k v k + n 2 k w ki, for all k 2 J. The approach described here is implemented in C. In order to handle the sparse matrix structure and sparse linear solvers, we use the software package PETSc [12, 13, 14]. 3.3 Failure of the Nodal Finite Element Approach With the weak formulation and numerical implementation described in 3.2, we observe that the approximate solutions fail to converge to the correct solution, specifically the numerical approximations converge to the wrong solution, both on domains with re-entrant corners and domains with curved boundaries. This is illustrated by analytical and numerical examples, shown later in

38 These seemingly surprising results motivate a careful re-examination of both the strong problem and the weak formulation. It turns out that the failures of convergence (to the correct solution) on domains with re-entrant corners and on domains with curved boundaries are due to di erent reasons. The former case is a result of insu cient regularity of the true solution, while the latter indicates the VPE with electric boundary conditions exhibits a form of Babuška paradox [1, 11] Insu cient Regularity of Exact Solution As shown in Lemma 3.1, the two VPEs (3.3a) and (3.4) are equivalent when the solution is in (H 2 ( )) d which guarantees the divergence is in H 1 ( ). Therefore, due to the fact that Laplace s equation with Dirichlet boundary condition has a unique solution in H 1 ( ), the equivalence follows. However, in general it may not necessarily hold that r u 2 H 1 ( ) andtheequation (3.5) could have a nonzero solution in a larger space, e.g., L 2 ( ). This is exactly what happens for domains with re-entrant corners. It is also pointed out in [66], that the assumption of Lemma 3.1 is satisfied for domains that are convex or with C 2 boundaries. However, this assumption is not satisfied if the domain has re-entrant (i.e., non-convex) corners. In such a case, the physically relevant (i.e., divergence-free) solution to (3.3) is not in H 1, while problem (3.4) possesses a solution in H 1, which, however, does not satisfy r u =inside. We can construct explicit examples to illustrate this. Let us assume that the domain R 2 has a re-entrant corner that is placed at the origin with boundaries lying on the positive x-axis and negative y-axis (see Fig. 3.1). For the sake of simplicity and convenience, we consider homogeneous boundary

39 27 y Ω O Ω x Ω Figure 3.1: Domain with re-entrant corner at the origin. condition with source function f =, and use polar coordinate system (r, ). u = in, (3.1a) n u = on@, (3.1b) r u = on@, (3.1c) Near the origin, the solution that is in (H 1 ( )) 2 with nonzero divergence can be characterized by u 1 =! u(r, ) = r 1/3 v(r, ) 1 sin 3 C A. cos 3 Clearly, the tangential component at the boundary near the origin is zero since u(r, ) = v(r, 3 /2) =. The divergence = r u 1 = 2 3 r 2/3 sin 2 3 vanishes at the boundary, is nonzero inside, andsatisfies rr + 1 r =.