DarcyLite: A Matlab Toolbox for Darcy Flow Computation

Transcription

1 Procedia Computer Science Volume 8, 216, Pages ICCS 216. Te International Conference on Computational Science DarcyLite: A Matlab Toolbox for Darcy Flow Computation Jiangguo Liu 1, Farra Sadre-Marandi 2, and Zuoran Wang 3 1 Department of Matematics, Colorado State University (USA), liu@mat.colostate.edu 2 Matematical Biosciences Institute, Oio State University (USA), sadre.1@mbi.osu.edu 3 Department of Matematics, Colorado State University (USA), wangz@mat.colostate.edu Abstract DarcyLite is a Matlab toolbox for numerical simulations of flow and transport in 2-dim porous media. Tis paper focuses on te finite element (F) metods and te corresponding code modules in DarcyLite for solving te Darcy equation. Specifically, four major types of finite element solvers, te continuous Galerkin (CG), te discontinuous Galerkin (DG), te weak Galerkin (WG), and te mixed finite element metods (MFM), are examined. Furtermore, overall design and implementation strategies in DarcyLite are discussed. Numerical results are included to demonstrate te usage and performance of tis toolbox. Keywords: Darcy flow, Flow in porous media, Mixed finite element metods, Weak Galerkin 1 Introduction Te Darcy s law is a fundamental equation for modeling flow in porous media. It is usually furter coupled wit transport equations. Two examples amongst te vast applications in tis regard are oil recovery in petroleum reservoirs [4, 1, 13] and drug delivery to tumors. fficient and robust Darcy solvers are needed for transport simulators [6]. Tere exist many finite element solvers for te Darcy equation, or more generally second order elliptic boundary value problems, e.g., te continuous Galerkin finite element metods [5], te discontinuous Galerkin finite element metods [2], te enanced Galerkin (G) finite element metods [13], te mixed finite element metods [7], and te newly developed weak Galerkin finite element metods [1, 16]. Tese types of finite element metods ave distinctive features in addition to certain common features. Different finite element solvers migt be favored by different users and/or applications. Teir implementations sare some common strategies but also need different treatments [11]. Based on tese considerations, we ave developed DarcyLite, a Matlab toolbox tat contains solvers for te Darcy equation and concentration transport equations. Matlab is cosen as te programming language, due to its popularity, easiness for coding, and integrated All tree autors were partially supported by US National Science Foundation Grant DMS Selection and peer-review under responsibility of te Scientific Programme Committee of ICCS 216 c Te Autors. Publised by lsevier B.V. 131 doi:116/j.procs

2 grapics functionalities. Tis toolbox is easy to use and efficiently solves medium-size 2-dim flow and transport problems. It is expandable. Tis allows incorporation of oter (new) types of solvers for flow and transport problems. But tis paper focuses on te Darcy solvers. In tis paper, we discuss mainly four finite element solvers (CG, DG, WG, MFM) and teir Matlab implementations in DarcyLite. Tis includes details on mes data structure, quadratures, finite element mass and stiffness matrices, andling of boundary conditions, assembly of global discrete linear systems, computation of numerical velocity and normal fluxes on edges, examination of local mass conservation, and presentation of numerical pressure and velocity. A numerical example is included to demonstrate te use and efficiency of tis toolbox. We consider 2-dim elliptic boundary value problems (Darcy problems) formulated as { ( K p) u = f, x Ω, p = p D, x Γ D, u n = u N, x Γ N (1), were Ω R 2 is a bounded polygonal domain, p is te primal unknown (pressure), K is a ydraulic conductivity (permeability) tensor tat is uniformly symmetric positive-definite, f is a source term, p D,u N are respectively Diriclet and Neumann boundary data, n te unit outward normal vector on Ω, wic as a nonoverlapping decomposition Γ D Γ N. We define a subspace and a manifold respectively for scalar-valued functions as follows H D, (Ω) = {p H 1 (Ω) : p Γ D =}, H D,pD (Ω) = {p H 1 (Ω) : p Γ D = p D }. Te variational form for te primal variable pressure reads as: Seek p HD,p 1 D (Ω) suc tat K p q = fq u N q, q HD,(Ω). 1 (2) Ω Ω Γ N Te Darcy equation can also be rewritten as a system of two first-order equations by considering te primal variable (pressure) and flux (velocity u = K p) as follows K 1 u + p =, u = f. (3) We define a subspace and a manifold respectively for vector-valued functions as follows H N, (div, Ω) = {v L 2 (Ω) 2 :divv L 2 (Ω), v Γ N = }, H N,uN (div, Ω) = {v L 2 (Ω) 2 :divv L 2 (Ω), v Γ N n = u N }. Te mixed variational formulation reads as: Seek u H N,uN (div, Ω) and p L 2 (Ω) suc tat (K 1 u) v p( v) = p D v n, v H N, (div, Ω), Ω Ω Γ D (4) ( u)q = fq, q L 2 (Ω). 132 Ω Ω For Darcy F solvers, two desired properties are (assuming u is te numerical velocity): Local mass conservation: For any element wit n being te outward unit normal vector on its boundary, tere olds u n = f. (5)

3 Normal flux continuity (in te integral form): For an interior edge sared by two elements i tat ave respectively outward unit normal vectors n i (i =1, 2), tere olds u 1 n 1 + u 2 n 2 =. (6) Tese important quantities are set as standard outputs of te F solvers in tis toolbox: (1) Numerical pressure average over eac element; (2) Numerical velocity at eac element center (and if needed, coefficients in te basis of te elementwise approximation subspace, e.g., RT or RT [] ); (3) Outward normal fluxes on all edges of eac element; (4) Local-mass-conservation residual on eac element if a F solver is not locally conservative; (5) Normal flux discrepancy across eac interior edge if te normal fluxes are not continuous. 2 F Solvers for Darcy quation: CG, DG, WG, MFM 2.1 Continuous Galerkin (CG) Te CG solvers for te Darcy equation are based on discretizations of te primal variational formulation (2). CG P 1 on triangular meses and CG Q 1 on rectangular meses are most frequently used. For a triangular mes T wit no anging nodes, let V be te space of all continuous piecewise linear polynomials on T.LetV = V H D, (Ω) and VD = V H D,pD (Ω) (after Lagrangian interpolation of Diriclet boundary data being performed). A finite element sceme using CG P 1 for te pressure is establised as: Seek p VD suc tat K p q = fq u N q, q V. (7) T T T T T After te numerical pressure p is obtained, te numerical velocity is calculated ad oc [4]. Te CG P 1 basis functions are actually barycentric coordinates. Computation and assembly of element stiffness matrices are relatively easy. CG P 1 or Q 1 scemesaveteleastnumbers of unknowns. But CG scemes are not locally mass-conservative and do not ave continuous normal fluxes across edges [8]. Post-processing is needed [5] to render a numerical velocity te above two properties. 2.2 Discontinuous Galerkin (DG) Let be a rectangular or triangular mes, define V k as te space of (discontinuous) piecewise polynomials wit a total degree k on. A DG sceme seeks p V k suc tat [13] A (p,q)=f(q), q V k, (8) were A (p,q)= + β Γ I ΓD K p q Γ N Γ I ΓD {K q n}[p ] + {K p n}[q] Γ I ΓD α [p ][q], (9) 133

4 F(q) = fq Γ N u N q + β Γ D K q np D + Γ D α p D q. (1) Here α > is a penalty factor for any edge Γ and β is a formulation parameter [13]. Depending on te coice of β, one ends up wit te symmetric interior penalty Galerkin (SIPG) for β = 1, te nonsymmetric interior penalty Galerkin (NIPG) for β = 1, and te incomplete interior penalty Galerkin (IIPG) for β =. Stability of te above DG sceme for a numerical pressure relies on te penalty factor. DG numerical velocity can be calculated ad oc. Te DG velocity is locally mass-conservative, but its normal component is not continuous across element boundaries. As pointed out in [2], tis leads to nonpysical oscillations, if te DG velocity is coupled to a DG transport solver in a straigtforward manner. Tis could make particle tracking difficult or impossible, if te DG velocity is used in a caracteristic-based transport solver. Te features and usefulness of te H(div) finite elements, especially, te BDM finite element spaces motivate post-processing of te DG velocity via projection [2]. Te post-processed velocity as te following properties [2]: (i) Te new numerical velocity as continuous normal components on element interfaces; (ii) Te new numerical velocity reproduces te averaged normal flux of te DG velocity; (iii) It as te same accuracy and convergence order as te original DG velocity. Details on implementation of te DG scemes and teir post-processing can be found in [2, 11]. 2.3 Weak Galerkin (WG) Te WG finite element metods are developed based on te innovative concepts of weak gradients and discrete weak gradients [16]. Instead of using te ad oc gradient of sape functions, discrete weak gradients are establised at te element level to provide nice approximations of te classical gradient in partial differential equations. Te WG approac considers a discrete sape function in two pieces: v = {v,v },te interior part v could be a polynomial of degree l inteinterior of an element, te boundary part v on could be a polynomial of degree m, its discrete weak gradient w,n v is specified in V (,n) P n () 2 (n ) via integration by parts: ( w,n v) w = v (w n) v ( w), w V (,n). For example, for a triangle T, WG(P,P,RT ) uses a constant basis function on T,aconstant basis function on eac edge on T, and specifies te discrete weak gradients of tese 4 basis functions in RT. Normalized basis functions for RT facilitate computation of tese discrete weak gradients [8, 1, 11], altoug small-size linear systems need to be solved in general. Te WG scemes for te Darcy equation rely on discretizations of (2). Let be a triangular or rectangular mes, l, m, n nonnegative integers, S (l, m) te space of discrete sape functions on tat ave polynomial degree l in element interior and degree m on edges, S (l, m) te subspace of functions in S (l, m) tatvanisonγ D. Seek p = {p,p } S (l, m) suc tat p Γ D = Q p D (projection of Diriclet boundary data) and 134 A (p,q)=f(q), q = {q,q } S (l, m), (11)

5 A (p,q):= K w,n p w,n q, F(q) := fq Γ N u N q. (12) After obtaining te numerical pressure p, one computes te WG numerical velocity by u = R ( K w,n p ), (13) were R is te local L 2 -projection onto V (,n).butitcanbeskippedwenk is a constant scalar on te element. Te normal fluxes are computed accordingly. Te WG scemes are locally conservative and produce continuous normal fluxes (in te integral form) [1, 11, 16]. 2.4 Mixed Finite lement Metods (MFM) Te mixed finite element scemes for te Darcy equation are based on discretizations of te mixed variational form (4). A pair of finite element spaces (V,W ) are cosen suc tat V H(div, Ω) for approximating velocity and W L 2 (Ω) for approximating pressure and togeter tey satisfy te inc-sup condition. Among te popular coices are (RT,P ), (BDM 1,P )for triangular meses and (RT [],Q ) for rectangular meses [1, 11]. For a rectangular or triangular mes, denote U = V H un,n (div; Ω) and V = V H,N (div; Ω). A mixed F sceme can be stated as: Seek u U and p W suc tat K 1 u v p ( v) = p D (v n), v V, Γ D ( u )q = (14) fq, q W. For implementation of (RT,P ), te edge-based basis functions for V can be used in computation and assembly of element matrices [1, 11]. A symmetric indefinite linear system is solved. Tis is taken care by Matlab backslas, altoug it is nontrivial beind te scene. An obvious advantage of te MFM scemes is te automatic local mass conservation (obtained by taking test function q = 1 in (14) 2nd equation) and normal flux continuity (built in te construction of H(div) finite element spaces). It is interesting to observe certain equivalence between te MFM scemes and te WG finite element scemes [1]. 3 Design and Implementation of DarcyLite DarcyLite is implemented in Matlab, wic is a popular interpretative language. Altoug it is different tan compilation languages like C/C++ or Fortran, Matlab is very efficient in matrix or array operations, since its core was implemented mainly in Fortran. Tis also means tat Matlab arrays are stored columnwise. Te integrated development environment and grapics functionalities offered by Matlab are very elpful for educational purposes. Te above observations guide our overall design of DarcyLite: Code modules and separation of tasks; Minimization of function calls; Columnwise storage and access for full matrices or arrays; 135

6 Utilizing Matlab (i, j, s)-format for sparse matrices; Performing operations on a mes for all elements simultaneously. Tis section furter elaborates on tese ideas troug a list of selected topics. Some data structures and implementation tecniques in DarcyLite sare te same spirit as tose in [1, 3]. 3.1 Mes Data Structure Mes information can be categorized as geometric (node coordinates, positions of element centers, etc.), topological (an element vs its nodes, adjacent edges of a given edge, etc.), and combinatorial (te number of neigboring nodes of a given node, etc.). Some basic mes data are used by all types of finite element solvers, even toug tey use some derived mes info in different ways. Mes data can be organized as (i) Primary info suc as te numbers of nodes/elements, node coordinates, an element vs its nodes: NumNds, Numms, node, elem (ii) Secondary info suc as an edge vs its vertices, an edge vs its neigboring element(s), an element vs its all edges: Numgs, edge, edge2elem, elem2edge, Leng, area, mcntr (iii) Tertiary info are more specific to particular finite element solvers. For instance, DG needs to know wat are te neigboring elements for a give element, MFM and WG need specific info on weter an edge is te 2nd edge of a given element. Te info on te (unit) normal vector of a given edge is also used by MFM and WG. For example, te geometric info about mes node coordinates is organized as an array of size NumNds*2, te topological info about triangular elements vs teir vertices is organized as an array of size Numms*3. Sown below is a code fragment for computing all triangle areas based on te columnwise storage in Matlab. k1 = TriMes.elem(:,1); k2 = TriMes.elem(:,2); k3 = TriMes.elem(:,3); x1 = TriMes.node(k1,1); x2 = TriMes.node(k2,1); x3 = TriMes.node(k3,1); y1 = TriMes.node(k1,2); y2 = TriMes.node(k2,2); y3 = TriMes.node(k3,2); TriMes.area = *abs((x2-x1).*(y3-y1)-(x3-x1).*(y2-y1)); Based on tese tree levels of mes info, DarcyLite organizes mes data as a structure tat as several dynamic fields. Te fields can be easily added or removed. Code modules are provided for mes info enricment (adding data fields at a iger level), see code modules TriMes_nric1.m, TriMes_nric3.m, RectMes_nric1.m, RectMes_nric3.m 3.2 Implementation of Quadratures Integrals on elements and element interfaces comprise a major portion of computation in finite element metods. Altoug tis looks like a simple issue, but code efficiency can be improved wen we adopt an unconventional approac. For example, in te WG(P,P,RT )scemefor te Darcy equation, one needs to compute te integrals f over all triangular elements. If a T K-point Gaussian quadrature is applied, ten te conventional approac would be K fdt T f(α k P 1 + β k P 2 + k P 3 )w k, T T, 136 T k=1

7 were (α k,β k, k ),w k (k =1,...,N) are respectively te barycentric coordinates and weigts of te quadrature points. Te order of summation and loop can be reversed: GlbRHS = zeros(dofs,1); NumQuadPts = size(gaussquad.trig,1); for k=1:numquadpts qp = GAUSSQUAD.TRIG(k,1) * TriMes.node(TriMes.elem(:,1),:)... + GAUSSQUAD.TRIG(k,2) * TriMes.node(TriMes.elem(:,2),:)... + GAUSSQUAD.TRIG(k,3) * TriMes.node(TriMes.elem(:,3),:); GlbRHS(1:Numms) = GlbRHS(1:Numms) + GAUSSQUAD.TRIG(k,4) * qnbc.fxnf(qp); end GlbRHS(1:Numms) = GlbRHS(1:Numms).* TriMes.area; 3.3 lement-level Small Matrices and Mes-level Sparse Matrices Finite element scemes involve computation of element-level small-size full matrices, e.g., mass matrices and stiffness matrices, and mes-level large-size sparse matrices, e.g., te global stiffness matrix. In general, we avoid using cell arrays of usual matrices, since tey may be inefficient. Instead we use 3-dim arrays. For example, in te WG(P,P,RT ) sceme for te Darcy equation, eac triangle involves one WG basis function for element interior and tree WG basis functions for te edges. Te discrete weak gradient of eac of te four WG basis functions is a linear combination of te tree RT normalized basis functions [1]. We organize tese coefficients as a tree-dimensional array CDWGB = zeros(trimes.numms, 4, 3); Te interaction of te discrete weak gradients of te 3 basis functions results in a 3-dim array ArrayGG = zeros(trimes.numms, 3, 3); Tese element-level small full matrices are assembled into te sparse global stiffness matrix DOFs = TriMes.Numms + TriMes.Numgs; GlbMat = sparse(dofs,dofs); Instead of using a loop over all elements, we utilize te (i, j, s)-structure in Matlab and te mes topological info: for i=1:3 II = TriMes.Numms + TriMes.elem2edge(:,i); for j=i:3 % Utilizing symmetry JJ = TriMes.Numms + TriMes.elem2edge(:,j); if (j==i) GlbMat = GlbMat + sparse(ii,ii,arraygg(:,i,i),dofs,dofs); else GlbMat = GlbMat + sparse(ii,jj,arraygg(:,i,j),dofs,dofs); GlbMat = GlbMat + sparse(jj,ii,arraygg(:,i,j),dofs,dofs); end end end See te following code modules for more details: Darcy_WG_TriPPRT_AsmSlv.m, Darcy_WG_RectQPRT_AsmSlv.m 137

8 3.4 nforcement of ssential Boundary Conditions For CG and WG, Diriclet boundary conditions are essential, but Neumann boundary conditions are natural, see quations (7,11). For MFM, Neumann conditions are essential but Diriclet conditions are natural, see quation (14). For DG, Diriclet conditions are enforced weakly [13] in te bilinear and linear forms, see quations (8,9,1). Teoretically, a zero essential boundary condition corresponds to a nice subspace for te numerical solution in te finite element sceme. For a non-zero essential condition, owever, we usually use te terminology manifold to accommodate te numerical solution. From te implementation viewpoint, to enforce essential boundary conditions, we need to modify te global sparse linear system obtained from a finite element sceme. Tere are basically two approaces, ard ( brutal ) and soft ( gentle ). We illustrate te latter using a simple linear system. Let {x 1,x 2 } be unknowns tat satisfy a linear system as follows [ ][ ] A11 A 12 x1 = A 21 A 22 x 2 [ b1 b 2 ]. Assume x 2 = c is te essential boundary condition. A brutal way is to modify te system to [ ][ ] [ ] A11 A 12 x1 b1 =. I x 2 c But tis usually damages te symmetry in te original linear system. An alternative is to consider solving A 11 x 1 = b 1 A 12 c, wic is a linear system wit a smaller size and maintains te symmetry of A 11 sown in te original system. Note tat [ ] [ ] [ ][ ] b1 A 12 c b1 A11 A = 12 b 2 A 21 A 22 c involves a simple matrix-vector multiplication (using te original global coefficient matrix) and we just need to take te first block in te final result vector. See te following code modules for more details: Darcy_CG_RectQ1t_AsmSlv.m, Darcy_WG_TriPPRT_AsmSlv.m, Darcy_WG_RectQPRT_AsmSlv.m 3.5 Interface wit Oter Software Packages DarcyLite focuses on solving flow and transport problems in te environment provided by Matlab. It can import triangular meses generated by oter packages, e.g., PD Toolbox and DistMes [12], wic also run on Matlab. DarcyLite provides functions for triangular mes conversion from te aforementioned two packages. Triangle is a popular triangular mes generator developed in C. DarcyLite can read in te mes data files generated by Triangle in te so-called flat file transfer (FFT) approac. 3.6 Grapical User Interface (GUI) of DarcyLite A grapical user interface (GUI) is provided wit DarcyLite for demonstration and education purposes. It was developed using Matlab built-in guide (grapical user interface development environment). Te GUI is based on te event-driven design principle and some tecniques in SNSAI [14] are adopted. Te GUI as tree parts as sown in Figure 1: Preparation, Numerical metod, and Presentation. 138

9 Figure 1: A screen snapsot of te Grapical User Interface (GUI) for DarcyLite I. Preparation. Tis part allows te user to specify details for a problem to be solved. (A) A domain is defined, including te endpoints along te x- and y-axis. Te default settings suggest a rectangular domain wit bot axes starting at and ending at 1. (B) Specify te Darcy equation tru a ydraulic conductivity K and a source term f. (C) Diriclet and Neumann boundary conditions are specified. Te two built-in options for K, f, and boundary conditions are defined according to two popular test examples. (D) A mes is to be generated. Te user can coose between a rectangular or triangular mes by specifying te numbers of uniform partitions nx, ny for te x, y-directions, respectively. Default settings suggest nx =2,ny = 2. Te Sow Mes button pops up a figure window for te user to ceck weter a correct mes as been generated. () Gaussian type quadratures (for edges, rectangles, and triangles) are cosen. Te default coices (5,25,13) are sufficient for most cases. II. Numerical Metod. Tis part of te GUI lists te four major types of finite element solvers. Te user can coose among Continuous Galerkin, Discontinuous Galerkin, Weak Galerkin, or Mixed Finite lement Metod. Ten te user must click te Run Code button. A popup will confirm tat te inputs from Part I are correct for te cosen finite element solver. Oterwise, an error message will pop up. III. Presentation. Te ceckboxes in tis part allow te user to display any combination of te numerical pressure and velocity profiles, local mass-conservation residual (LMCR), and flux discrepancy across edges. 139

10 4 A Transport Solver: Implicit uler + Weak Galerkin DarcyLite contains also finite element solvers for transport problems in 2-dim prototyped as c t + (vc D c) =f(x, y, t), (x, y) Ω,t (,T), c(x, y, t) =, (x, y) Ω,t (,T), (15) c(x, y, ) = c (x, y), (x, y) Ω. Here c(x, y, t) is te unknown (solute) concentration, v te Darcy velocity, D>adiffusion constant, f a source/sink term. Tere exist various types of finite element metods for transport problems. Here we briefly discuss a numerical sceme tat utilizes te implicit uler for timemarcing and weak Galerkin for spatial approximation. For simplicity, we assume Ω is a rectangular domain equipped wit a rectangular mes and a numerical velocity u as been obtained from applying a finite element solver tat is locally mass-conservative and as continuous normal fluxes. Te unknown concentration is approximated using te lowest order finite element space WG(Q,P,RT [] ). Let C (n) (n 1) be suc an approximation at discrete time t n, ten tere olds for n 1, (C (n),w) Δt (u C (n), w,dw) +ΔtD ( w,d C (n), w,dw) = (C (n 1),w) +Δt (16) (f,w). An initial approximant C () can be obtained via local L 2 -projection of c (x, y) intotewg finite element space. Te Implicit uler + Weak Galerkin sceme as two nice properties: (i) C (n) represents intuitively te cell average of concentration on any element. (ii)tescemeislocally and ence globally conservative. Tis is verified by taking a test function w tat as value 1 in one element interior but in all oters and on all edges. Te 2nd term on te left side of te above equation caracterizes interaction of te flow (Darcy velocity) and te concentration (discrete weak) gradient. Te 3rd term is a symmetric term similar to tat in any elliptic problem. Te 1st term on te rigt side represents te mass at te previous time moment, wereas te last term depicts te source/sink contribution during te time period [t n 1,t n ]. 5 Numerical xperiments Tis section presents numerical results to demonstrate te use of DarcyLite. We consider an example of coupled flow and transport. A numerical Darcy velocity is fed into te transport solver discussed in te previous section. For te Darcy equation, Ω = (, 1) 2, te ydraulic conductivity profile is taken from [7] (alsousedin[1, 11]). Tere is no source. A Diriclet condition p = 1 is specified for te left boundary and p = for te rigt boundary. A zero Neumann condition is set for te lower and upper boundaries. Te Darcy solver WG(Q,P,RT [] ) is used on a uniform 1 1 rectangular mes (but for grapics clarity, Figure 2 Panel (a) sows results of 4 4 mes). 131

11 For te transport problem (15), T =, D =1 3, tere is no source. An initial constant concentration is placed in [, ] [, ]. Te RT [] numerical velocity from te Darcy solver is used in te I+WG sceme (16) wit a 1 1 rectangular mes and Δt = Figure 2 Panel (b)(c)(d) present numerical concentration profiles for time moments t =, 5,, respectively. It can be observed tat te transport pattern reflects well te flow features: (i) Te upper part of te domain as stronger flow and ence te concentration front in te upper part advances faster tan tat in te lower part. (ii) In Panel (d), in te middle of te domain, an almost vertical transport pat can be observed. Tis clearly reflects te strong flow in te domain from position (, ) downward to position (, ), see Panel (a) also. Note tat te mass flux on te domain boundary is zero. It can be cecked numerically tat te total mass in te domain C (n) remains at for all discrete time moments (t =trougt = witincrement.5). Tis verifies te mass conservation property (ii) discussed in Section 4. However, tere are small negative concentrations. liminating oscillations in numerical concentrations of convection-dominated transport problems is a nontrivial issue [9]. Particular treatments for te Implicit uler + Weak Galerkin transport solver are currently under our investigation. WG: Numerical pressure and velocity Projected initial concentration.8 1 (a) (b) time t = Numerical concentration Numerical concentration.8 1 (c) time t =5 (d) time t = 1.8 Figure 2: xample: Coupled Darcy flow and transport. (a) Numerical pressure and velocity. (b) Initial concentration. (c) Concentration at time t =5. (d) Concentration at time t =. Results for (c)(d) are obtained using WG(Q,P,RT [] )( =1 2 ) and implicit uler Δt =

12 6 Concluding Remarks DarcyLite is a small-size code package developed in Matlab for solving 2-dim flow and transport equations. It will be extended to include 3-dim solvers and test cases. For te flow equation, DarcyLite provides four major types of F solvers (CG, DG, WG, MFM) on triangular and rectangular meses. CG post-processing [5] and te enanced Galerkin (G) [13] will be included later. For te transport equation, DarcyLite provides solvers for bot steady-state and transient problems. For te latter, it offers solvers of ulerian type and ulerian-lagrangian type [15]. For coupled flow and transport problems, solvers for a 2-pase model [8] are provided. DarcyLite is being extended to include more solvers tat are efficient, robust, and respect pysical properties, e.g., local conservation, positivity-preserving. Besides applications like petroleum reservoir and groundwater simulations, DarcyLite is being extended to include flow and transport problems in biological media, e.g., drug delivery. TeURLfortiscodepackageis ttp:// References [1] J. Alberty, C. Carstensen, and S. Funken. Remarks around 5 lines of matlab: sort finite element implementation. Numer. Algor., 2: , [2] P. Bastian and B. Riviere. Superconvergence and (div) projection for discontinuous galerkin metods. Int. J. Numer. Met. Fluids, 42: , 23. [3] L. Cen. ifem: an integrated finite element metods package in matlab. Tec. Report, Mat Dept., Univ. of California at Irvine (29), 29. [4] Z. Cen, G. Huan, and Y. Ma. Computational metods for multipase flows in porous media. SIAM, 26. [5] B. Cockburn, J. Gopalakrisnan, and H. Wang. Locally conservative fluxes for te continuous galerkin metod. SIAM J. Numer. Anal., 45: , 27. [6] C. Dawson, S. Sun, and M. Weeler. Compatible algoritms for coupled flow and transport. Comput. Met. Appl. Mec. ngrg., 193: , 24. [7] L. Durlofsky. Accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities. Water Resour. Res., 3: , [8] V. Ginting, G. Lin, and J. Liu. On application of te weak galerkin finite element metod to a two-pase model for subsurface flow. J. Sci. Comput., 66: , 216. [9] V. Jon and. Scmeyer. Finite element metods for time-dependent convection diffusion reaction equations wit small diffusion. Comput. Met. Appl. Mec. ngrg., 198: , 28. [1] G. Lin, J. Liu, L. Mu, and X. Ye. Weak galerkin finite element metdos for darcy flow: Anistropy and eterogeneity. J. Comput. Pys., 276: , 214. [11] G. Lin, J. Liu, and F. Sadre-Marandi. A comparative study on te weak galerkin, discontinuous galerkin, and mixed finite element metods. J. Comput. Appl. Mat., 273: , 215. [12] P. Persson and G. Strang. A simple mes generator in matlab. SIAM Review, 46: , 24. [13] S. Sun and J. Liu. A locally conservative finite element metod based on piecewise constant enricment of te continuous galerkin metod. SIAM J. Sci. Comput., 31: , 29. [14] S. Tavener and M. Mikucki. Sensai: A matlab package for sensitivity analysis. ttp:// tavener/fscu/snsai/sensai.stml. [15] H. Wang, H.K. Dale, R.. wing, M.S. spedal, R.C. Sarpley, and S. Man. An ellam sceme for advection-diffusion equations in two dimensions. SIAM J. Sci. Comput., 2: , [16] J. Wang and X. Ye. A weak galerkin finite element metod for second order elliptic problems. J. Comput. Appl. Mat., 241:13 115,