Adaptive Finite Element Methods in Electrochemistry

Transcription

1 10666 Langmuir 2006, 22, Adaptive Finite Element Methods in Electrochemistry David J. Gavaghan,* Kathryn Gillow, and Endre Süli Oxford UniVersity Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K. ReceiVed April 28, In Final Form: August 7, 2006 In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique. 1. Introduction Microelectrodes are widely used for a variety of electroanalysis and electrochemical measurement techniques. Because in general there is no direct means of relating the measured quantity (current, potential, etc.) to the underlying chemical system of interest, all such techniques are underpinned by mathematical models that are used to relate these output measurements to the input and chemical parameters of interest (applied potential, concentrations, partial pressures, reaction rates, etc.). The mathematical models take the form of reaction-convection-diffusion equations, coupled with boundary conditions describing the particular electrochemical control technique. Modern microdevices allow the analysis of very fast reactions and extend clinical use to in-vivo measurements. For these devices, mathematical models are analytically intractable. As we outline below, numerical solution is complicated by boundary singularities, the complexity of the chemical processes, and, for clinical devices, by semipermeable membranes used to prevent surface spoiling (e.g., by protein deposition). As a result, previous numerical approaches (see ref 1 for a comprehensive review) have had difficulty demonstrating accuracy for general problems. The long-term goal of our research has therefore been to develop a general, and preferably completely automated, approach that will overcome all of these difficulties and generate approximations for the quantity of interestsin our case, generally the current flowing in an electrochemical cellsto a user-prescribed tolerance. In this article, we describe our work to date on the development of adaptive finite element methods (using both continuous and discontinuous approaches, to be described later) for amperometric techniques at a variety of microelectrode geometries and configurations. All of this work uses the technique of deriving an a posteriori error bound to drive the adaptivity, which is somewhat technical and involved from a mathematical viewpoint. In this review of our work to date, we therefore describe in detail only our work on steady-state problems, giving only a brief overview of how we have extended this work to timedependent problems. Technical mathematical details are given only for the simplest model problem in the Appendix, with readers Part of the Electrochemistry special issue. * Corresponding author. (1) Alden, J. A. D. Phil. Thesis, University of Oxford, Oxford, U.K., referred to earlier work for details of the underpinning mathematics for the more complex examples. We begin this work with a brief review of previous approaches to the mathematical modeling and numerical simulation of electrochemical processes at microelectrodes, together with a brief overview of the use of adaptive numerical techniques in electrochemistry by other authors. 2. Previous Work Diffusion processes at microelectrodes are typically modeled in two dimensions by utilizing an inherent symmetry in the electrode geometry. However, the small size of the electrode (typically in the range of µm) and the typical time span of the experiments (the range from the transient out to the steady state) mean that an account must be taken both of the concentration gradient normal to the electrode surface (as for a large planar electrode) and the concentration gradient parallel to the electrode surface. This parallel component results in a nonuniform current distribution over the electrode surface that is usually termed the edge effect, and it is this effect that is usually quoted as making the microdisk problem challenging (Michael et al. 2 ). In practice, the edge effect means that the standard finite difference approaches on uniform spatial meshes that have been used for 1D electrochemistry simulation problems since they were introduced by Feldberg in a seminal paper in have very slow rates of convergence and thus a prohibitively large number of unknowns is required to compute an accurate solution. In general, previous approaches that have attempted to overcome these problems fall into five categories: 1. (approximate) analytical solutions; 2. special integration technique at the electrode edge to allow for increased current; 3. matching a locally valid series solution near the electrode edge to the far-field numerical solution; 4. conformal maps that either increase the number of points near the singularity or effectively remove the singularity; and 5. mesh refinement to put more points near the singularity. (2) Michael, A. C.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1989, 267, 33. (3) Feldberg, S. W. Anal. Chem. 1964, 36, /la061158l CCC: $ American Chemical Society Published on Web 09/27/2006

2 AdaptiVe Finite Element Methods in Electrochemistry Langmuir, Vol. 22, No. 25, Approximate Analytical Solutions. The only problem for which a known simple closed-form analytical solution exists is that of steady-state diffusion to a microdisk electrode. This was first presented by Saito in and further simplified by Crank and Furzeland in As a result, it has become a very common test problem for numerical methods. However, approximate analytical solutions have been developed for a variety of other problems including chronoamperometry at microband 6-9 and microdisc electrodes, linear sweep voltammetry at microband 14 and microdisc 15 electrodes, steady-state E, ECE, and DISP1 reactions at channel microband electrodes, and EC reactions at inlaid and recessed microdisk electrodes. 21,22 Although these solutions may not be exact, the authors are often able to give an indication of their relative accuracy, so these solutions are very useful for comparison with numerical simulations. More recently, Mahon and Oldham 23 revisited the problem of finding the transient current at the disk electrode under diffusion control and derived two overlapping short- and long-time polynomial functions that are shown to give excellent agreement with all previous work Allowing for Increased Current. This approach was developed by Heinze for chronoamperometry 24 and cyclic voltammetry 25 at microdisk electrodes. It uses a special method of flux integration at the edge of the electrode. However, this method appears to be very problem-dependent, and thus it seems unlikely that it could provide the basis for a general simulation algorithm. In addition, no attempt is made to correct the effect that the boundary singularity has on the concentration values Local Series Solutions. This approach was first used by Crank and Furzeland 5 for steady-state diffusion to a microdisk electrode and was subsequently adopted by Gavaghan and Rollett 26 for the corresponding time-dependent problem and further developed for use with the finite element method by Galceran et al. 27 The idea is to find a series solution that is valid in the neighborhood of the singularity and to use a finite difference scheme further away. By comparing the number of nodes required (4) Saito, Y. ReV. Polarogr. (Jpn.) 1968, 15, 177. (5) Crank, J.; Furzeland, R. M. J. Inst. Math. Its Appl. 1977, 20, 355. (6) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 230, (7) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 225, (8) Oldham, K. B. J. Electroanal. Chem. 1981, 122, (9) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, M. J. Electroanal. Chem. 1987, 217, (10) Aoki, K.; Osteryoung, J. J. Electroanal. Chem. 1981, 122, 19. (11) Aoki, K.; Osteryoung, J. J. Electroanal. Chem. 1984, 160, 335. (12) Rajendran, L.; Sangaranarayanan, M. V. J. Electroanal. Chem. 1995, 392, (13) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1982, 140, 237. (14) Aoki, K.; Tokuda, K. J. Electroanal. Chem. 1987, 237, (15) Aoki, K.; Akimoto, K.; Tokuda, K.; Matsuda, H.; Osteryoung, J. J. Electroanal. Chem. 1984, 171, 219. (16) Ackerberg, R. C.; Patel, R. D.; Gupta, S. K. J. Fluid Mech. 1977, 86, 49. (17) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 217, 33. (18) Leslie, W. M.; Alden, J. A.; Compton, R. G.; Silk, T. J. Phys. Chem. B 1996, 100, (19) Levich, V. G. Physiochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, (20) Newman, J. Electroanal. Chem. 1973, 6, 187. (21) Galceran, J.; Taylor, S. L.; Bartlett, P. N. J. Electroanal. Chem. 1999, 466, 15. (22) Galceran, J.; Taylor, S. L.; Bartlett, P. N. J. Electroanal. Chem. 1999, 476, 132. (23) Mahon, P. J.; Oldham, K. B. Electrochim. Acta 2004, 49, (24) Heinze, J. J. Electroanal. Chem. 1981, 124, 73. (25) Heinze, J. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, (26) Gavaghan, D. J.; Rollett, J. S. J. Electroanal. Chem. 1990, 295, 1. (27) Galceran, J.; Gavaghan, D. J.; Rollett, J. S. J. Electroanal. Chem. 1995, 394, 17. to achieve a given accuracy in the current, it is clear that using the series solution is much more efficient than a standard finite difference method on a regular mesh Conformal Mapping Techniques.The first type of map, as proposed by Taylor et al., 28 is designed to increase the density of points in the neighborhood of the singularity. Essentially the map leads to a mesh that expands in both spatial directions. However, as Taylor et al. note, a poorly chosen grid expansion factor may be counterproductive and lead to large errors. In addition, it is not obvious how to choose the factors. The second type, as used by Michael et al., 2 Verbrugge and Baker, 29 and Amatore and Fosset, 30 essentially folds the radial axis at the electrode edge, which effectively removes the singularity. All of the maps lead to a problem in the transformed coordinates that are solved using finite difference schemes on a regular mesh. The map used by Michael et al. is essentially a transformation to oblate spherical coordinates, and this was used successfully to study cyclic voltammetry at microdisk electrodes. They also showed that the map could be used for the simulation of coupled chemical steps by applying it to an EC mechanism. This map was also used by Deakin et al. 31 for cyclic voltammetry at microband electrodes and by Lavagnini et al. 32 for the simulation of voltammetric curves at microdisk electrodes. Verbrugge and Baker 29 extended this work. They used a transformation that maps the infinite strip produced in ref 2 into a rectangle. They then used these new coordinates to study chronoamperometry at a microdisk electrode. The conformal map proposed by Amatore and Fosset 30 has the advantage of producing a finite rectangle in the transformed space. This was used to study steady-state and near-steady-state diffusion at microdisk electrodes. The advantage of these transformations to a finite domain is that the far-field boundary conditions may be applied exactly. This cannot be done in an infinite domain. Instead the far-field conditions have to be applied at a sufficient distance from the electrode surface in the hope that the truncated problem will accurately approximate the true problem. Alden and Compton 34 also use the map described in ref 30 to generate working curves for ECE, DISP1, EC 2 E, DISP2, and EC mechanisms at microdisk and spherical/hemispherical electrodes. In addition, they have compared this map to the one in ref 29, and they claim that it is superior in that it is more efficient for diffusion-only processes and the matrices resulting from a finite difference discretization are better conditioned. More recently, Oleinick, working with Amatore and Svir, 33 used a quasi-conformal mapping technique to derive what are claimed to be the most accurate and efficient simulations to date for diffusion at microdisk electrodes. These authors also compare their approach to that of Amatore and Fosset. The main drawback to the conformal map approach is that the equations in the transformed coordinates are far more complex than those in the original coordinates and deriving the transformed partial differential equation may be very complicated (and must be done afresh for each new system of equations that is tackled). In addition, as observed by Taylor et al., 28 poorly chosen (28) Taylor, G.; Girault, H. H.; McAleer, J. J. Electroanal. Chem. 1990, 293, 19. (29) Verbrugge, M. W.; Baker, D. R. J. Phys. Chem. B 1992, 96, (30) Amatore, C. A.; Fosset, B. J. Electroanal. Chem. 1992, 328, 21. (31) Deakin, M. R.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1986, 215, (32) Lavagnini, I.; Pastore, P.; Magno, F.; Amatore, C. A. J. Electroanal. Chem. 1991, 316, 37. (33) Oleinick, A.; Amatore, C.; Svir, I. Electrochem. Commun. 2004, 6, 588. (34) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 9606.

3 10668 Langmuir, Vol. 22, No. 25, 2006 GaVaghan et al. conformal maps may be counterproductive, and different maps may perform well for different problems, making the choice of conformal map difficult. In addition, there is no control over the error in the subsequent simulations, and it is gaining such control that leads to the adaptive mesh refinement approach that is the primary focus of this article Mesh Refinement. As we will show below, these problems can be overcome by using a mesh refinement approach within the numerical method. This approach is standard for the numerical simulation of 1D problems in electrochemistry, with Feldberg (with various co-workers) again having introduced the key concepts into the electrochemical literature This approach was used in two dimensions by Gavaghan for steady-state diffusion, chronoamperometry, and linear sweep voltammetry at microdisk electrodes and by Bartlett and Taylor 41 for recessed microdisk electrodes. The idea is to use a finer mesh where the error is large, for example, at the boundary singularity at the electrode edge for inlaid disks and along the electrode surface and at the corner singularity for recessed disks. The mesh then expands until it is very coarse at large distances from these points. Alden and Compton have also used expanding meshes for studying reactions at channel microband electrodes. These authors use uniform meshes parallel to the electrode surface but exponentially expanding meshes perpendicular to it. However, this still required over one million mesh points for an accurate approximation of the current. Even more mesh points were needed for low flow rates; indeed for very low flow rates these simulations did not provide accurate approximations of the current because of the excessive memory requirements. That these methods can give very accurate solutions for both the microdisk and microband has been confirmed in two excellent recent review papers by Britz and co-workers. 42,43 These papers compare the results from the three primary approaches (analytic, conformal map, and mesh refinement) to give a set of reference values for each of these problems that are accurate to better than 0.1% at all times in both cases. These papers provide an excellent overview and reference to all previous work on these problems. However, the drawback with all of these early attempts at using mesh refinement is that the refinement strategy is entirely heuristic and often relies upon knowledge of the analytic solution or an accurate analytic approximation to assess the accuracy and convergence of the numerical method. This is also costly given the slow convergence. In either case, there is no reliable gauge of how accurate the solution is on a given (refined) mesh nor of whether the mesh is in any way optimal. Clearly, this is a rather ad hoc approach and requires the inefficient intervention of the user. Our aim when we began the work reviewed in this article was therefore to produce accurate and efficient algorithms to approximate the current using the finite element method. This has allowed us throughout our subsequent work to build a rigorous mathematical analysis that can be used to generate meshes upon which we can approximate the quantity of interest, in our case, the current, to within a prescribed tolerance, eliminating the need for excessive convergence testing or the ad hoc mesh generation approach described above. (35) Feldberg, Stephen W. J. Electroanal. Chem. 1981, 127, 1. (36) Feldberg, Stephen W.; Goldstein, Charles I. J. Electroanal. Chem. 1995, 397, 1. (37) Mocak, J.; Feldberg, S. W. J. Electroanal. Chem. 1994, 378, 31. (38) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, (39) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, (40) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, (41) Bartlett, P. N.; Taylor, S. L. J. Electroanal. Chem. 1998, 453, 49. (42) Britz, D.; Poulsen, K.; Strutwolf, J. Electrochim. Acta 2004, 50, 107. (43) Britz, D.; Poulsen, K.; Strutwolf, J. Electrochim. Acta 2005, 51, 333. Other authors have considered using an adaptive finite element method approach in electrochemistry. Perhaps the most comprehensive body of work has been conducted by Bieniasz using a patch-adaptive approach. (See a series of papers beginning with ref 44 to the most recent, ref 45.) This work covers a huge range of electrochemical application problems but to date is restricted to one spatial dimension. In two spatial dimensions, Nann and Heinze 46,47 used a gradient-based approach to drive the adaptivity, but this again requires a heuristic approach to determining when convergence has been achieved. Abercrombie and Denuault 48 built upon our early work to consider alternative electrode geometries and also introduced a patch recovery technique that has the potential to improve convergence rates and accuracy in some instances. 3. Example Problems Considered In this article, we will consider the generic applicability of our approach by solving four model problems that each illustrate its ability to overcome a particular numerical difficulty encountered by previous authors. First, the simplest possible problem of finding the steady-state current to a microdisk electrode (and for which we have a closed-form analytical solution) allows us to give a comprehensive description of the mathematics behind the approach (although the technical details are left for the Appendix) and to illustrate how the adaptive algorithm automatically generates meshes that provide increasingly accurate solutions to the problem. The problem of a steady-state solution of a firstorder EC reaction (catalytic reaction mechanism) at a recessed microdisk has been chosen because by varying the governing reaction rate the reaction layer can be confined within the recess or can spread beyond the recess and be affected by the numerical problems arising from the corner singularity at the mouth of the recess. The third problem considered, that of the steady-state current at a channel microband electrode, allows us to illustrate the use of the method in Cartesian (rather than cylindrical) coordinates and to extend the treatment to a range of more complex reaction mechanisms. These three problems are each treated using the continuous Galerkin finite element method, but as we show for the third problem (the channel microband electrode), the more recently developed discontinuous Galerkin finite element method (DG- FEM) has strong advantages for certain problems (in this case, avoiding the need to stabilize the numerical approach in convection-dominated problems). The advantages of the DGFEM approach become even more apparent in the fourth model problem that we considerscalculating the steady-state current at a membrane-covered Clarke electrode. We end this article by presenting some previously unpublished work that illustrates how it is possible to utilize the fact that in calculating the a posteriori error bound we must calculate a higherorder solution to the dual problem. This allows us to derive an alternative formulation for the current in terms of this higherorder solution, providing large improvements to the accuracy of the current calculation at no extra cost. Finally, we give a brief overview of how we have extended this work to time-dependent problems. (44) Bieniasz, L. K. J. Electroanal. Chem. 2000, 481, 115. (45) Bieniasz, L. K. J. Electroanal. Chem. 2004, 565, 273. (46) Nann, T.; Heinze, J. Electrochem. Commun. 1999, 1, 289. (47) Nann, T.; Heinze, J. Electrochim. Acta 2003, 48, (48) Abercrombie, S. C. B.; Denuault, G. Electrochem. Commun. 2003, 5, (49) Østerby, O. Technical Report DAIMI PB-534, Department of Computer Science, University of Aarhus, Ny Munkegade, Bldg. 540, DK-8000 Aarhus C, Denmark, 1998.

4 AdaptiVe Finite Element Methods in Electrochemistry Langmuir, Vol. 22, No. 25, Simplest Model Problem We begin our review with the first problem that we considered and the one that is probably the simplest possible problem of this typesdetermining the steady-state current at a microdisk electrode. Because this problem has a closed-form analytic solution, 4 it makes an ideal model problem, and we give a full description of our approach here, including the technical derivations of the necessary a posteriori bounds 83 in the Appendix. We consider the simple reaction mechanism A ( ne - ) B (1) at a disk electrode. By assuming semi-infinite mass transport and utilizing the axial symmetry of the problem, the governing steady-state diffusion equation can be written in cylindrical polar coordinates as 2 u r + 1 u 2 r r + 2 u z ) 0 (2) 2 with u ) c/c 0, where c 0 is the bulk concentration of species A. Here, the spatial coordinates have been normalized with respect to the electrode radius a Boundary Conditions. Assuming fast enough kinetics to ensure zero concentration on the electrode surface, the boundary conditions are u ) 0 r e 1 z ) 0 u n ) 0 r > 1 z ) 0 r ) 0 z g 0 (3) u ) 1 r, z f The boundary conditions on z ) 0 ensure that there is a boundary singularity at (1, 0) where u/ r is discontinuous, causing problems for standard numerical techniques. We also note that currently we are required to solve the problem on a semi-infinite domain (0, ) (0, ). However, to solve the problem numerically we must use a finite region [0, r max ] [0, z max ]. Because the analytical solution for the concentration field is known for this problem (see below), we may test our numerical algorithm by choosing any values for r max and z max and using the exact solution as the boundary condition on these parts of the boundary. Most of our numerical results are produced using this approach. Alternatively, we could choose large values of r max and z max and apply the boundary condition u ) 1, and we shall also present results using this approach Current. The quantity of interest in this problem is the nondimensional current at the electrode surface given by I ) π 2 0 1( u r dr ) z) I (4) z)0 4FDc 0 a where I is the dimensional current Exact Solution. An exact solution to this problem was given by Saito 4 in 1968 u(r, z) ) 1-2 sin m π 0 m J 0 (rm)e-zm dm (5) with J 0 being the zeroth-order Bessel function. Differentiating this and substituting into eq 4 gives the familiar value for the nondimensional current of I ) 1. Crank and Furzeland 5 gave an alternative form of this solution as u ) sin-1{ 2 {1 - π 2 z 2 + (1 + r) 2 + z 2 + (1 - r) 2} z > e r e 1, z ) π sin-1 { 1 r} r > 1, z ) 0 which is more convenient when comparing to numerical approximations. We also use this form of the solution as the boundary condition on r ) r max and z ) z max Numerical Techniques. Our numerical technique is based on the finite element method (FEM) and is described in detail in the Appendix. Use of the FEM involves obtaining the numerical solution of the governing partial differential equations by subdividing the domain into triangles and obtaining piecewise linear approximations to that solution on each triangle. Our primary interest is in calculating an estimate of the current, which we will denote by I est, to within a user-specified tolerance, so we require methods of estimating the error in the numerical approximation. If the exact value of the current is denoted by I, then the usual standard theoretical bounds on the error I - I est can be derived without the need to find I est first. These are known as a priori error bounds and indicate the expected order of convergence of the solution. Unfortunately, these bounds are usually of the form I - I est e Ch k, where h is the mesh size (or the largest mesh size on a nonuniform mesh), C is some constant that is dependent on the exact solution of the partial differential equation, and k depends on both the degree of the approximating polynomial and the regularity of the true solution u to the PDE. Such bounds are useful in that they show the rate at which the method converges as h f 0; however, in general the size of the constant C is not known, and hence the bound is not computable. All of our work therefore makes use of a posteriori error bounds, which are much more useful because they can be expressed in terms of the finite element solution u h in such a way that they are computable and can therefore be used as the basis of an adaptive mesh refinement strategy. As we show below, this then ensures that the quantity of interest (here, the current) is both reliably and efficiently computed. The details of how such an a posteriori bound is calculated is rather technical and involved and is therefore given only in the Appendix. There we also outline how this bound can be incorporated into an adaptive finite element algorithm that ensures the calculation of the current to within a user-defined error tolerance with near-optimal levels of computational effort Results. Using the adaptive algorithm described in the Appendix, a series of computational meshes is derived upon which we must calculate the solution and the a posteriori error bound. The algorithm terminates when the error bound is less than TOL, a user-defined tolerance, and at this point, the numerical estimate of the current has an error of at most TOL. In this example, we choose a value of TOL ) 0.05; that is, we request at most a 5% error in the current. As we show (and as might be expected), in practice we obtain greater accuracy than this because the a posteriori error bound overestimates the error (in this case, by a factor of about 3.) Taking r max ) z max ) 2 to define the solution region (and using eq 6 to define the solution on r max and z max ), we choose to begin with a regular mesh with 32 triangles. (Note that we are utilizing axial symmetry to render the problem in 2D cylindrical coordinates and that this cylindricity is built into the governing equation, eq 2.) In Table 1, we show the number of triangles (n_tri), the number of nodes (n_nods), currents, and error estimates on the resulting sequence of meshes that is generated by our (6)

5 10670 Langmuir, Vol. 22, No. 25, 2006 GaVaghan et al. adaptive algorithm. The effectivity index is the estimated error divided by the actual error and gives a measure of the sharpness of the error bound. (Clearly, an ideal algorithm will result in an index value of 1, although in practice it will always be greater than 1.) It is only because we know the exact value of the current in this case that the effectivity index is computable. The results of this process allow us to calculate the current to our specified accuracy on a remarkably coarse mesh; the sixth and final mesh in Table 1 actually gives a value of the current that is accurate to almost 1% using only 391 unknowns and minimal CPU time. The meshes generated by the algorithm are given in Figure 1. It is immediately clear that our algorithm is able to determine very quickly (by the third mesh) that the numerical errors are caused by the boundary singularity, and successive meshes home in on this area, successively reducing the level of error in the overall calculation of the current until the required tolerance is attained Comparison with Regular Meshes. Just how much better this approach is can be illustrated by comparing the adaptive approach to the use of a simple regular mesh constructed by dividing the domain into N equal spacings in each coordinate direction and triangulating by joining the top left corner of each square to the bottom right. We can also use this comparison to show the superiority of the approximation N ψ (u h ) (cf. eq 55 in the Appendix) for the current. We use a regular mesh and approximate the current using N ψ (u h ): as can be seen, halving the mesh spacing roughly halves the error in the current (Table 2), suggesting that the error in the current is order O(h). This is backed up by the a priori error bound derived in ref 51. A similar level of accuracy to that of the adaptive algorithm (an error of around 1%) requires around 100 mesh spacings in each direction, a total of nodes altogether, and over 30 times the CPU time. The standard method of calculating the current used in electrochemistry is simply to insert the numerical solution u h into the original definition of the current; that is, the current is given by The part of the final mesh closest to the electrode surface is shown in Figure 2 and has the same characteristics as the previous meshes. 5. Recessed Disk Electrode The power of the adaptive finite element technique described above is particularly clear when we consider our second model problem of the recessed disk electrode. We consider the EC reaction mechanism (catalytic reaction mechanism) for which we have a means of obtaining an analytic solution and are able to show that our method can give good accuracy on relatively coarse meshes because it picks out those features of the problem that cause numerical difficulties and refines the mesh only in those areas Theory. We consider the case of S being present in a large excess so that we can assume pseudo-first-order kinetics, 22,52,53 allowing the EC mechanism to be described by A ( ne - f B (8) B + S 98 k A + Y where k is the pseudo-first-order reaction rate, n is the number of electrons involved, and S and Y are electroinactive species. Here we have assumed that the equilibrium constant for the reaction is so large that the reverse process can be ignored. The geometry of the problem is shown in Figure 3. Following refs 22 and 53, a nondimensional concentration u can be defined as u ) c B c B s s where c B is the concentration of species B and c B is the concentration of B on the electrode surface. Normalizing the spatial coordinates with respect to the electrode radius allows the radius to be taken to be equal to 1 in all simulations. The concentration, u, then satisfies (9) π 2 0 1( u h z ) r dr (7) z)0 with - 2 u + Ku ) 0 (10) instead of eq 55. In Table 3, we show that this approximation converges to the true value, N ψ (u), at less than half the rate of N ψ (u h ); estimating the error between eq 7 and the exact value of the current to be O(h 0.4 ) means that we would require over nodes using eq 7 to achieve the same accuracy as the sixth adaptive mesh gave with only 391 nodes using eq 55. It is therefore very clear that it is better to estimate the current using the expression N ψ (u h ) defined by eq Results for Large r max and z max. Finally, we present the results obtained from the adaptive algorithm by choosing r max ) z max ) 80 and applying the boundary condition u ) 1 on these parts of the boundary. In Table 4, we show the details of the resulting sequence of meshes that is generated by our adaptive algorithm. Again we see that relatively few nodes are needed to estimate the current accurately. The erratic values of the effectivity index on the finer meshes are due to the fact that the exact solution of the problem we are solving is slightly larger than unity (because we are applying a boundary condition that is slightly larger than it should be). (50) Gavaghan, D. J. J. Electroanal. Chem. 1997, 420, 147. (51) Harriman, K.; Gavaghan, D. J.; Houston, P.; Süli, E. Electrochem. Commun. 2000, 2, 157. K ) ka2 D B (11) being the dimensionless reaction rate and a being the actual electrode radius. D B is the diffusion coefficient of B. The boundary conditions of the problem are then u ) 1 r e 1 z ) 0 u f 0 r, z f u n ) 0 r ) 0 z > 0 (12) r ) 1 0 e z e L r > 1 z ) L with the normalized current given by I )- π 2 0 1( u r dr (13) z) z)0 The numerical results that we obtain will be compared to the exact solution and approximate analytical solutions of Galceran

6 AdaptiVe Finite Element Methods in Electrochemistry Langmuir, Vol. 22, No. 25, Figure 1. Meshes produced by an adaptive finite element method for the steady-state disk electrode problem. et al., 22 which hold assuming that pseudo-first-order conditions are maintained. (52) Denuault, G.; Fleischmann, M.; Pletcher, D.; Tutty, O. R. J. Electroanal. Chem. 1990, 280, Results. To illustrate the power of the method in picking out the salient features of the problem, we consider only the case (53) Rajendran, L.; Sangaranarayanan, M. V. J. Phys. Chem. B 1999, 103, 1518.

7 10672 Langmuir, Vol. 22, No. 25, 2006 GaVaghan et al. Table 1. Triangulations Produced by the Adaptive Finite Element Method for the Steady-State Disk Electrode Problem mesh n_tri n_nods estimated current actual error estimated error effectivity index Table 2. Values of the Numerical Approximation N ψ(u h)tothe Current N ψ(u) and the Error N ψ(u) - N ψ(u h) Evaluated on a Regular Mesh with N Mesh Spacings in Each Direction N current error order Figure 2. Part of the final mesh closest to the electrode surface. Table 3. Values of the Approximation to the Current Using Equation 7, the Associated Error, and Its Order of Convergence Evaluated on a Sequence of Regular Meshes with N Mesh Spacings in Each Direction N current error order Table 4. Triangulations Produced by the Adaptive Finite Element Method for the Steady-State Disk Electrode Problem on the Domain [0, 80] [0, 80] mesh n_tri n_nods estimated current actual error estimated error effectivity index of L ) 0.5 and compare with the analytical results for K ) 1 and The adaptive tolerance for each case is chosen to guarantee 1% accuracy in the estimated current. The final mesh for each of the problems is shown in Figure 4, and these look as we would expect. For the lower K value, diffusion dominates, and the corner singularity causes a problem when evaluating the current; consequently, more nodes are needed near the corner. For the higher K value, the reaction takes place almost entirely within the recess, so the mesh may remain coarse further away. In fact, we see that the mesh size above the electrode surface is proportional to the thickness of the reaction layer, which is known to be K -1/2. These ideas are shown more clearly in Figure 5, which depicts the contours of the solution for each value of K. It is clear that our completely automated mesh generation algorithm is able to pick out the features in the solution that will have an impact on the simulated current as the governing parameters (in this case, K and L) are varied and will refine the mesh appropriately. The numerical values of the current, error bounds, and CPU time are shown in Table 5. Agreement is again excellent. We have again achieved much better accuracy than the tolerance that we prescribed, particularly so for K. 1, because the upper bound on the error in the computed current overestimates to an extent that increases with K. Figure 3. Geometry of a recessed microdisk electrode. 6. More Complex Reaction Mechanisms at a Channel Microband Electrode Our third example application is the consideration of more complex reaction mechanisms at a channel microband electrode. The channel microband electrode has become an increasingly popular alternative to the microdisk electrode in electroanalytical work, particularly for mechanistic studies. 54 Channel microband electrodes are hydrodynamic electrodes in which the working electrode is embedded in a wall of a rectangular duct or pipe. Forced convection arises because of the movement of electrolyte solution that is pumped through the duct. 55 If laminar flow is established in the channel, then the velocity profile is parabolic (Poiseuille flow) with the maximum velocity occurring at the center of the channel as shown in Figure 6. This problem therefore extends the range of applicability of our techniques to incorporate convection, across a wide range of flow rates, from diffusiondominated to the more numerically challenging convectiondominated case. We also extend the range of kinetics investigated by considering an EC j E reaction mechanism for the two cases j ) 1, 2 with the reactions given by A + e f B k j jb 98 C (14) C + e f E where k j is the reaction rate. Assuming equal diffusion coefficients (54) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, (55) Fisher, A. C. Electrode Dynamics; Oxford Chemistry Primers, No. 34; Oxford University Press: Oxford, U.K., 1996.

8 AdaptiVe Finite Element Methods in Electrochemistry Langmuir, Vol. 22, No. 25, Figure 4. Final meshes for calculating the current to a recessed microdisk electrode during an EC mechanism with small and large reaction rates. Figure 5. Contours of the solution for the two different K values. There are 15 equally spaced contours between u ) 0.05 and D for species A, B, and C and laminar or Poiseulle flow with a velocity profile in the x direction ν x ) ν 0 (1 - (h - y) 2 /h 2 ), where h is half the channel height, we can nondimensionalize using the approach given in ref 1 to obtain the steady-state transport equations for nondimensional concentrations a, b, and c as with dimensionless rate constant - 2 a + ν a ) 0 (15) - 2 b + ν b + jk ECj E bj ) 0 (16) - 2 c + ν c ) K ECj E bj (17) K ECj E ) k j x 2 j - 1 e c D, (18) and ν ) ( 1 / 2 P s y(2 - p 1 y), 0) where P s is the shear rate Peclet K Table 5. Steady-State Current to a Recessed Microdisk Electrode for L ) 0.5 and Different Values of K adaptive tolerance number that measures the relative importance of convection and diffusion and we define exact current N ψ(u h) a posteriori error bound no. of nodes in final mesh CPU time (s) p 1 ) x e h P s ) 3 2 p 12 p 2 (19) p 2 ) 4hν 0 3D (20) where x e is the dimensional electrode width as shown in Figure 7.

9 10674 Langmuir, Vol. 22, No. 25, 2006 GaVaghan et al. Figure 6. Parabolic flow profile in the channel microband electrode. The flow is assumed to be uniform in the z direction. Figure 7. Coordinates for the channel microband electrode. The boundary conditions are given by a n ) b n ) c n ) 0 on y ) 0, x > 1, and x < 0 a n ) b n ) c n ) 0 The nondimensional current at reaction rate K ECjE is given by where u ) (a, b, c) and the primary quantity of interest is N eff, the effective number of electrons transferred, where and J 0 (u) is the current for the case of simple electron transfer (i.e., K ) 0). Note that we have implicitly made the standard assumption here that the dimensions of the band electrodes are such that the band can be considered in effect to be infinitely long in the z direction, allowing symmetry to be assumed and reduction of the solution space to two (x and y) dimensions Simplifying the Problem. This problem can be simplified by defining d ) a + b to obtain with boundary conditions on y ) 2 p 1, - < x < a n, b n, c n f 0 as x f,0< y < 2 p 1 (21) a ) c ) 0, b n )- a n on y ) 0, 0 e x e 1 a f 1, b, c f 0 as x f - J K (u) ) 0 1[( a + y) y)0 ( c y) y)0 ] dx (22) N eff ) J K (u) J 0 (u) (23) - 2 d + ν d + jk ECj E (d - a)j ) 0 (24) d f 1 as x f - (25) d n ) 0 on the rest of Ω (26) If we now solve for a, d, and c, then the problems are now coupled only through the partial differential equations (not the boundary conditions) and so can be solved sequentially Streamline Diffusion Stabilization. As suggested earlier, conventional numerical methods applied to convection-dominated diffusion equations may exhibit numerical instabilities because of the failure of the numerical method to conform to the necessary discrete maximum principle (typically manifested by oscillatory solutions) unless severe restrictions are imposed on the mesh size. This can be overcome by use of the streamline-diffusion finite element method (SDFEM), more details of which are given in refs 60 and Results. In previous work, 61 we have compared the results of our algorithms to the work of previous authors across the range of realistic values of the governing parameters P s and K (the dimensionless rate constant given by K ) P -2/3 s K ECjE), and the interested reader is referred to these papers. Here we wish only to illustrate again that our approach can automatically pick out the salient features of the problem that will affect the numerical accuracy of the quantity of interest (in this case, N eff ). Our previous work 61 took as an example the case with h ) 0.02 cm, D ) cm 2 s -1, and x e ) cm with four values for log P s, one corresponding to a low flow rate (log P s )-2.4), two to a medium flow rate (log P s )-1.2 and log P s ) 0), and one to a high flow rate (log P s ) 2) and with K chosen to range between and The tolerance chosen for our adaptive algorithm ensured 2.5% accuracy in the current or equivalently 5% accuracy in N eff. In all cases, excellent agreement was obtained with previous results 54 for both cases j ) 1, 2. However, in all cases we were able to solve the problem using between 2 and 3 orders of magnitude fewer unknowns than previous authors used. How this is achieved by our algorithm is clear by considering some examples of the final meshes produced by the adaptive algorithm for the EC 2 E mechanism as shown in Figure 8. Only the parts of the meshes closest to the electrode surface are shown because it is here that most of the refinement takes place. It is clear that the algorithm is able to pick out the parameters that affect the current (here, P s and K) and refine the mesh appropriately so that for the two diffusion-dominated cases mesh points are packed in around the singularities at the edges of the electrode but for the cases where convection becomes more prominent mesh points are packed at the singularities and also right along the electrode surface. 7. Discontinuous Galerkin Finite Element Methods In all of the previous examples, we have used the continuous finite element method within our adaptive algorithm. However, this algorithm performed only h refinement so that the mesh size was changed or refined to improve accuracy, rather than p refinement where the degree of the polynomial within the finite element approximation is changed (i.e., instead of using linear basis functions over the elements, higher-order functions (quadratics, cubics, etc.) are used). An increasingly popular alternative is to combine both h and p refinement within a discontinuous Galerkin finite element framework. The discontinuous Galerkin finite element method is ideal for hp refinement because interelement continuity is imposed only in a weak sense so that the degree of the polynomial may vary freely from element to element. This is beneficial because in regions where the solution (56) Hughes, T. J. R.; Brooks, A. In Finite Elements in Fluids; Gallagher, R. H., Norrie, D. H., Oden, J. T., Zienkiewicz, O. C., Eds.; John Wiley & Sons: Chichester, U.K., 1982; Vol. 4, p 47. (57) Johnson, C.; Nävert, U.; Pitkäranta, J. Comput. Methods Appl. Mech. Eng. 1984, 45, 285. (58) Johnson, C. Numerical Solution of Partial Differential Equations by the Finite Element Method; Cambridge University Press: Cambridge, U.K., (59) Hansbo, P.; Johnson, C. Streamline Diffusion Finite Element Methods for Fluid Flow; von Karman Institute Lectures; von Karman Institute for Fluid Dynamics: Rhode Saint Genèse, Belgium, (60) Harriman, K.; Gavaghan, D. J.; Houston, P.; Süli, E. Electrochem. Commun. 2000, 2, 567. (61) Harriman, K.; Gavaghan, D. J.; Houston, P.; Kay, D.; Süli, E. Electrochem. Commun. 2000, 2, 576.

10 AdaptiVe Finite Element Methods in Electrochemistry Langmuir, Vol. 22, No. 25, Figure 8. Samples of the final meshes produced by the adaptive finite element algorithm. is well-behaved (i.e., the solution and its derivatives are defined and square integrable) a greater increase in the accuracy of the numerical solution per number of degrees of freedom may be obtained by keeping the mesh size fixed and increasing the degree of the approximating polynomial. This is known as p refinement. Of course in the neighborhood of a singularity (such as the boundary singularity occurring at the edge of the electrode), mesh refinement is still needed to confine the effects of the singularity to as small a region as possible. This suggests that an adaptive algorithm that combines h and p refinement should be a very efficient way of proceeding. A further advantage is that it is now straightforward to use rectangular rather than triangular finite elements because the difficulty of adaptively refining a rectangular mesh without introducing hanging nodes (which is not permissible with the continuous Galerkin formulation) is removed. A corollary is that the problem of ensuring that triangular elements do not become poorly shaped (long and thin) when adaptively refining a mesh is also removed. Additionally, the discontinuous Galerkin finite element method has a key advantage over the standard continuous finite element method for this particular problem, namely, that there is no need for streamline diffusion stabilization when dealing with convection-dominated diffusion problems. 62 This means that reactiondiffusion and reaction-convection-diffusion problems may be treated in a uniform manner. To solve a problem using DGFEM, we first rewrite the partial differential equation in its weak formulation. As in the continuous finite element method, this consists of multiplying the partial differential equation by a test function, integrating over all elements and applying Green s theorem. However, interelement continuity and the boundary conditions are then applied in a weak sense so that the actual DGFEM solution is not continuous across interelement boundaries and does not precisely satisfy the boundary conditions. A detailed description of how to derive the weak formulation for a reaction-diffusion equation is given in ref 63 and in our earlier paper. 64 Here we give two examples of the utility of the DGFEM approach for electrochemical problems. First, we give a brief (62) Houston, P.; Schwab, Ch.; Süli, E. SIAM J. Numer. Anal. 2002, 39, description of some results and meshes obtained for the EC j E reaction mechanism at a channel microband electrode. Second, we consider its application to the Clarke membrane-covered electrode Example 1: EC j E Reaction Mechanism. Again we consider an EC j E reaction mechanism at a channel microband electrode for j ) 1, 2. The equation and boundary conditions and model parameter values for the problem are the same as those used in section 6, with the tolerance for our adaptive algorithm again chosen to obtain 2.5% accuracy in the current or equivalently 5% accuracy in N eff. The key result of these computations was that the number of degrees of freedom required was significantly less than was required using the continuous adaptive algorithm. For the diffusion-dominated problems, the number of degrees of freedom using the hp DGFEM algorithm was about 1 / 5 of the number required by the continuous algorithm, whereas for convection-dominated problems the improvement was even greater because only about 1 / 10 of the number of degrees of freedom was required to achieve a given tolerance. Some example meshes close to the electrode surface are given in Figure 9 for the EC 2 E reaction mechanism. Again we see that extra mesh points are needed near the boundary singularities, but along the electrode surface and away from the electrode the degree of the polynomial is increased, which leads to fewer degrees of freedom in the whole mesh Example 2: Clark Electrode. Our second example of DGFEM is the Clark electrode, which was first developed in 1956 by Leland Clark 65 as a means of measuring blood oxygen tension (P O2 ). It remains the primary means of monitoring blood P O2 in clinical medicine, and its use has spread to areas as diverse as sewage treatment, soil chemistry, and the production of wine and beer. This electrode assembly is immersed in aqueous electrolyte solution and protected from poisoning by a tightly stretched plastic membrane that is permeable to oxygen. The (63) Oden, J. T.; Babuška, I.; Baumann, C. E. J. Comput. Phys. 1998, 146, 491. (64) Harriman, K.; Gavaghan, D. J.; Süli, E. Technical Report NA04/19, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, (65) Clark, L. C. Trans. Am. Soc. Internal Organs 1956, 2,

11 10676 Langmuir, Vol. 22, No. 25, 2006 GaVaghan et al. Figure 9. Samples of the final meshes produced by the hp-adaptive finite element algorithm. reason that we have chosen this as our fourth example now becomes clear: we must solve the diffusion equation both in the presence of a boundary singularity at the electrode edge and in the presence of an internal interface within the solution region at which derivatives of the solution are not continuous. (Note that because the fluxes across the material interfaces are continuous this can be straightforwardly overcome using a finitevolume-type approach. 66 We include this example here as an example of the dramatic improvements that can be achieved by using the discontinuous rather than the continuous approach with the finite element method.) The simplest form of the chemical reaction that takes place at the cathode when oxygen is reduced is O 2 + 2H 2 O + 4e - ) 4OH - (27) Assuming that the potential difference between the two electrodes is sufficiently large that all oxygen reaching the cathode is reduced, then the current is directly proportional to the oxygen concentration, and we have the basis of a measuring device Mathematical Model. Assuming that the cathode is a disk-shaped electrode of radius r c (typically ranging from 10 to 100 µm) embedded in an infinite planar insulating material, then the problem effectively reduces to that described in section 4 but with the added complication of the protective membrane and the electrolyte layer, with symmetry yielding the 2D solution region illustrated in Figure 10. Above the cathode, we have three layers with the following properties: 1. an electrolyte layer (e) of thickness z e (typically between 2 and 10 µm); 2. a membrane layer (m) of thickness z m (typically between 5 and 25 µm), where the membrane is porous to oxygen but typically has a diffusion coefficient that is at least an order of magnitude lower than that of the electrolyte; and 3. a layer of sample (s) taken to be infinitely thick, where the diffusion coefficient is assumed to be of the same order of magnitude as that of the electrolyte. (66) Versteeg, H. K.; Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method; Addison-Wesley: Reading, MA, Figure 10. Schematic representation of the geometry for the mathematical model of the Clark electrode. Rather than solving for the concentration of oxygen that is discontinuous between the three layers, we consider instead the partial pressure of oxygen p (related to the concentration of oxygen c by Henry s law: c )R k p where k ) e, m, s and R k is the solubility in the given layer), which is governed by the cylindrical diffusion equation R k D k( 1 r r( r p r) + 2 p z 2) ) 0 (28) in each layer. (Note that R k D k is the permeability.) The current is then given by r I ) 2πnFP e 0 c( p r dr (29) z) z)0 where F is the Faraday constant and n is the number of electrons per molecule transferred in the reduction reaction. Because we are assuming that the reaction is of the form given in eq 27, we take n ) 4. To nondimensionalize the equations, we set p ) p/p 0, where p 0 is the bulk partial pressure, and rˆ ) r/r c and ẑ ) z/r c, where

12 AdaptiVe Finite Element Methods in Electrochemistry Langmuir, Vol. 22, No. 25, r c is the electrode radius. Dropping the hats gives the dimensionless equations P k( 1 r r( r p r ) + 2 p z ) ) 0 (30) 2 in each layer where P k is the relevant dimensionless permeability. We assume that oxygen depletion takes place in all three layers but also that there exist (dimensionless) distances r max and z max at which no further oxygen depletion takes place so that the problem may be solved in the finite domain [0, r max ] [0, z max ]. In our numerical simulations, we have chosen r max and z max to be the same and with values of between 40 and 80, with the precise value depending on the values of the other parameters in the problem Boundary and Internal Conditions. The assumption that all oxygen reaching the cathode is reduced gives the boundary condition together with p ) 0 on z ) 0 0e r e 1 (31) p z ) 0 on z ) 0 1< r < r max (32) p r ) 0 on r ) 0 0< z < z max (33) p ) 1 r ) r max,0< z < z max z ) z max, < r < r max (34) For this problem, the layers also introduce internal conditions that must be satisfied at the material interfaces, and because the partial pressure is continuous, we have p z)zē ) p z)ze + (35) p z)zm - ) p z)zm + (36) Here, z e and z m are the dimensionless values of z at the interfaces, and z ē and z + e are defined by z e ( ) lim sf0 (z e ( s 2 ) The expressions z m - and z m + are defined analogously. We also impose the continuity of the oxygen fluxes across the material interfaces via P e( p z ) ) P z)ze - m( p z ) z)ze + P m( p z ) ) P z)zm - s( p z ) z)zm + (37) (38) Note that this final requirement results in discontinuities in p / z across the material interfaces, and again this is the reason that this particular test problem has been chosen as a good illustration of the power of the adaptive DGFEM algorithm Current. The dimensionless current is given by I ) π 2 0 1( p z ) r dr (39) z)0 An exact expression for the analytical current is given by eq 20 of ref 67. In eq 22 of ref 67, Galceran et al. also give an approximate analytical expression for the current that is valid for membranes that are not too thick and not too impermeable (i.e., for parameter regimes in which neither z m. z e nor ɛ 1,ɛ 2 f 1 holds). We shall compare our numerical approximations to each of these expressions, where ɛ 1 and ɛ 2 are the dimensionless parameters ɛ 1 ) P e - P m ) P e /P m - 1 P e + P m P e /P m + 1 ɛ 2 ) P s - P m ) P s /P m - 1 P s + P m P s /P m + 1 (40) (41) Numerical Results. We illustrate the accuracy and effectiveness of our algorithm as a function of the dimensionless parameters ɛ 1 and ɛ 2. In Figure 11a, we choose z e ) 0.5 and z m ) 1 and ɛ 1 ) ɛ 2 ) , corresponding to values of P e /P m ) P s /P m ) (i.e., P e and P s are an order of magnitude larger than P m ). The change in the gradient of the normalized partial pressure can be clearly seen at the material interfaces. In Figure 11b, we choose ɛ 1 ) ɛ 2 ) 0.1, giving P e /P m ) P s /P m ) 1.22 so that P e, P m, and P s are all of the same order of magnitude. In this case, it is much harder to see the change in the gradient across the material interfaces. In Table 6, we compare the current values obtained by our hp-adaptive discontinuous Galerkin finite element algorithm with those given by the analytical solution and the approximate analytical solution in eqs 20 and 22, respectively, of ref 67. In each case, we have chosen the adaptive tolerance to be 1% of the approximate solution. Clearly, we obtain very good agreement in all cases. To illustrate the adaptive algorithm at work, we consider the problem with z e ) 0.5, z m ) 1, and ɛ 1 ) ɛ 2 ) , and we set the tolerance to be 0.5% of the exact current. Part of the final mesh is shown in Figure 12; beyond the illustrated part of the domain, the mesh is coarse and regular and the degree of the polynomial is 1. The automated algorithm has resulted in a mesh that concentrates extra nodes around the boundary singularity at (1, 0) where the electrode meets the insulator, with extra nodes also added at the material interfaces. Where refinement is needed away from these singularities, it takes the form of p refinement although the maximum degree of the polynomial does not exceed 3. The whole mesh has a total of 1837 elements and 7633 degrees of freedom. This is a considerable improvement on the adaptive algorithm with continuous piecewise linear basis functions that requires more than degrees of freedom to solve the problem with the same parameters. Solving this problem on a regular mesh would therefore prove prohibitively expensive in terms of computational effort. 8. Utilizing the Higher-Order Dual Solution Because the adaptive algorithm requires us to compute the piecewise quadratic finite element approximation to the dual problem, we expect to be able to obtain more accurate current values by using this solution rather than a piecewise linear solution. To utilize this more accurate solution for the simplest model problem, we follow the approach described in ref 68. This is easily extended to other problems. We write u ) u 0 + g, where u 0 H 1 E 0(Ω) and g H 1 E (Ω). We recall from eq 53 that the current may be found by evaluating (67) Galceran, J.; Salvador, J.; Puy, J.; Cecilia, J.; Gavaghan, D. J. J. Electroanal. Chem. 1997, 440, 1. (68) Giles, M.; Technical Report NA97/11, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, 1997.

13 10678 Langmuir, Vol. 22, No. 25, 2006 GaVaghan et al. Figure 11. Typical solutions for the normalized partial pressure. Figure 12. Part of the final mesh produced by the discontinuous adaptive algorithm. Table 6. Comparison of Numerical Approximation to the Current Using DGFEM with the Exact and Approximate Currents Given by Equations 20 and 22 of Reference 67 for a Range of Parameter Values z e z m ɛ 1 ) ɛ 2 ref 67 eq 20 eq 22 error ref 67 I h bound B(u, V) for any V H ψ 1 (Ω). In particular, we may choose V) w where w is the solution of the dual problem given in eq 56. Thus, we have I ) B(u, w) ) B(u 0 + g, w) (42) ) B(u 0, w) + B(g, w) (43) Now u 0 H 1 E 0(Ω), so the weak formulation of the dual problem (eq 56) tells us that B(u 0, w) ) 0. Hence, the current may also be written as B(g, w) g H 1 E (Ω). This definition may be shown to be independent of the choice of g H 1 E (Ω), and so, as we did for the primal, we suppress g and write this alternative formulation of the current as J(w). We may base a numerical approximation of the current on this by evaluating J(w h ), where w h is the piecewise quadratic finite element approximation to w calculated by the adaptive algorithm Convergence of Numerical Approximations of the Current. The optimal rate of convergence of N ψ (u h )ton ψ (u) achieved by smooth solutions is second-order convergence. However, it can be shown that if the mesh is designed appropriately then second-order convergence may be achieved Table 7. Order of Accuracy of the Approximations to the Current on Adaptive Meshes TOL n_nods N ψ(u h) error order J(w h) error order for the model problem; see, for example, Schwab 69 page 96. (Here, convergence is measured in terms of ( n_nods) -1 where n_nods is the number of nodes in a given triangulation.) Thus, for our adaptive algorithm to be competitive it should achieve second-order convergence. In Table 7, we see that the optimal second-order accuracy is achieved by the adaptive algorithm. We also observe that the approximation J(w h ) has second-order convergence and that the approximations are much more accurate than those given by N ψ (u h ). However, we note that we have no bound for the error in the current computed using J(w h ) and so in the absence of an exact solution we have no gauge of the accuracy of this approximation. 9. Extension to Time-Dependent Problems Each of the above approaches to a posteriori error-bound driven adaptive FEM can be extended to time-dependent problems, as described in refs This approach still requires that we solve the dual problem numerically to compute the bounds that drive the adaptivity. However, because a time- (69) Schwab, Ch. p- and hp-finite Element Methods: Theory and Applications in Solid and Fluid Mechanics; Clarendon Press: Oxford, U.K., (70) Harriman, K.; Gavaghan, D. J.; Süli, E. Electrochem. Commun. 2003, 5, 519. (71) Harriman, K.; Gavaghan, D. J.; Süli, E. J. Electroanal. Chem. 2004, 569, 35. (72) Harriman, K.; Gavaghan, D. J.; Süli, E. J. Electroanal. Chem. 2004, 573, 169.