It is simple, and also much appreciated because it respects a discrete version of the maximum principle; in particular, the approximate solution does not oscillate and stays within its physical bounds. The convergence analysis of the FVM for diffusion equations in this particular case is thoroughly performed in Eymard et al. The error analysis relies on the conservativity and consistency of the numerical flux.
Unfortunately, all the linear schemes devised in the general case of unstructured meshes and general linear operators do not seem to satisfy a discrete version of the maximum principle; for severely distorted meshes or for highly anisotropic operators, the risk of oscillations and violation of the physical bounds is high. Ongoing research using in particular nonlinear schemes for linear problems is under way to avoid these drawbacks see the proceedings of Finite Volume for Complex Applications V.
Finite volume method
Nevertheless, the fluxes must be carefully approximated in order to ensure the stability and convergence of the scheme. Provided that, for an explicit scheme, the so-called CFL condition be satisfied this condition prescribes a bound for the time step with respect to the size of the mesh , these schemes satisfy the following properties see Eymard et al. The Godunov scheme is a very efficient scheme when the above solution of the Riemann problem is easily constructed.
When the solution of the Riemann problem becomes too complex or expensive, alternative schemes based on approximate Riemann solvers can be used see Eymard et al. See also these references for some extensions to multi-dimensional problems. Eymard, T. Finite volume methods. Ciarlet and J. North-Holland, Amsterdam, Eymard and J. Finite volumes for complex applications V. ISTE, London, Feistauer, J. Felcman, and I.
Mathematical and computational methods for compressible flow. Numerical Mathematics and Scientific Computation. The trace reconstruction operator is 26 such that the final flux reads Within this section, we present hybrid and mimetic schemes. Here, the discrete solution space is given as 14 , such that additional face unknowns are introduced. Here, the fluxes are given, for all , as 28 with the face unknowns , the cell unknowns u K , and coefficients.
These coefficients are chosen such that the resulting scheme is coercive and consistent. Defining the matrices, 29 we obtain the cell flux vector as 30 with , , and e is the vector with entries equal to one. Additionally, let be the conormal matrix, and let be the matrix that contains the scaled distance vectors: Then, the consistency condition is given by From this consistency condition, we can derive a general form of the coefficient matrix : 33 with a stabilization matrix such that. Therefore, the matrix is symmetric if the stabilization matrix is chosen symmetric.
A simple choice of is 36 with. This results in the final matrix For this matrix, the consistency condition 32 is fulfilled by construction, whereas the coercivity can be proven by using some kind of stability condition, see [ 33 ]. Another interesting scheme that fits into this framework, and also fits into the recently developed gradient discretization framework [ 15 ], is the Hybrid Finite-Volume HFV scheme [ 8 ]. The main idea of this scheme is the construction of a consistent discrete gradient, which at the same time defines the form , such that the consistency is naturally fulfilled.
In the following we shortly introduce the scheme and present the main ideas. Let us recall the weak formulation of problem 1 : Find such that. Therefore, the scheme is defined by the discrete gradient reconstruction operator. First, let us define a discrete gradient on each cell as 39 the consistency of this formula follows thanks to However, to end up in a coercive form, we need an additional stabilization term, which is defined as Inserting these discrete gradients into the form 38 and reordering of terms lead to the form that is defined in Thus, the fluxes and the coefficient matrix 30 can also be identified.
Therefore, the hybrid finite-volume scheme also belongs to the family of hybrid mimetic schemes. The coefficient matrix, a detailed description, and the proof of consistency and coercivity can be found in [ 8 ]. Remark 4. For a more detailed summary of finite-volume schemes we refer to [ 34 , 35 ]. In this chapter, the behavior of the different finite-volume schemes, that have been presented in the last section, is investigated.
Detailed explanations can be found in [ 21 , 24 ]. The mimetic finite-difference scheme with fluxes 30 and local cell matrix 37 is denoted as MFD scheme. The hybrid finite-volume scheme defined by the discrete gradients 41 is denoted as HFV scheme. The Box method [ 38 , 39 ] is a vertex-centered finite-volume scheme that uses finite-element basis functions on each cell to calculate fluxes over sub-control volume faces.
All simulations are performed using the open-source simulator DuMu x [ 40 ], which comes in the form of an additional DUNE module [ 41 ]. In this section, more general boundary conditions are considered, where the Dirichlet conditions are taken into account using the function such that the weak solution has to satisfy where denotes the trace operator. A discrete seminorm on the space is given by 42 which is a norm on the space.
Using the Cauchy-Schwarz inequality, one can show that see [ 8 ] for more details 44 with Owing to inequality 44 , the discrete norm 43 will also be used for the hybrid and mimetic schemes, although it is not a discrete norm for these schemes since it does not depend on the face unknowns. For measuring the coercivity of the schemes, the following estimate is defined In the next section the coercivity estimate is evaluated for the numerical solution.
Please note that this is not sufficient to show the coercivity of the schemes, but it serves as a good indicator. In the following test cases, the properties that have been mentioned in Section 2. Hereby, the numerical and exact solutions are denoted as u n where n indicates the grid refinement level and , respectively.
Remark 5. In this section, hexahedral grids are considered or quadrilateral grids for the case of a two-dimensional domain. Such grids are, in general, not admissible in the sense of Definition 1, because the faces are usually non-planar. For more details, we refer to [ 24 , 37 , 42 ].
In [ 42 ], the authors suggest to introduce, for strongly curved faces, additional degrees of freedom accounting for tangential velocities. These additional velocities are related to the planar face which is defined through the averaged normal vector. Such additional degrees of freedom are not introduced here, because we did not observe any convergence rate reduction for the considered examples.
Within this section, the convergence rates of the different schemes are compared for three-dimensional test problems with smooth solutions. The convergence of the family of schemes 19 has been proven, under the assumption of coercivity, in [ 21 ]. Here, we demonstrate that the convergence rates are similar to well-established schemes.
These examples are based on our previous work [ 24 ], but here we are using different norms and a more challenging tensor for the highly anisotropic test case. The convergence behavior is investigated for the meshes shown in Figure 1. For all grids, we use the dune-alugrid module [ 45 ].
Except for the checkerboard mesh, all grids exhibit non-planar faces. Therefore, we calculate the integrated normal vectors, see [ 24 ]. The first test case is similar to test 1 from the FVCA6 benchmarks [ 43 ]. The permeability tensor is The discrete L 2 -errors and H 1 -errors are shown in Figure 2. From top to bottom: checkerboard, random, non-convex, curved mesh. It can be seen that for increasing mesh refinement, the schemes converge with second order accuracy in the L 2 -norm and at least first order accuracy in the H 1 -norm.
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The convergence rates of all schemes are quite similar. We also observe that the MFD scheme produces the smallest errors. Considering Table 1 , we observe that the coercivity estimates are bounded from below which indicates that all schemes are coercive for this test case on all grids.
Table 1 Coercivity estimates and number of Newton iterations NIt for convergence test case one using the grids shown in Figure 1. Figure 3 shows the errors for test case two, with solution As in the previous example, no results could be obtained for the MPFA-O scheme on the checkerboard mesh due to the non-conformity of the grid.
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Considering Table 2 , we observe that the coercivity estimates are bounded from below which indicates that all schemes are coercive also for this test case. In addition, for the checkerboard mesh there are some cells and faces where some of the coefficients in the sub-flux definition 20 are negative. Table 2 Coercivity estimates and number of Newton Iterations NIt for convergence test case two using the grids shown in Figure 1. Remark 6. For the above test cases, the classical linear TPFA method does not converge. This is well-known for non-K-orthogonal grids, see for example [ 6 ].
Therefore, the TPFA scheme has not been considered so far. Within this section, the linearity preservation property of the schemes is investigated, which is a good indicator for consistency of schemes. This example is based on our recent work [ 21 ]. The considered domain and the grid are shown in Figure 4 right.
The domain consists of two sub-domains and. The transition from to is located at , and the tensors are chosen as Dirichlet conditions are set at the domain boundary, equal to the exact solution modified after [ 21 ]. The exact solutions in the sub-domains are Figure 4 left depicts the exact solution.
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Please note that the exact solution and the corresponding flux function are globally continuous within the domain. It can also be seen that the grid is non-matching at the transition of the sub-domains. Such non-matching grids often occur in faulted geological environments. To read in the grid, the opm-grid module from the Open Porous Media OPM initiative 2 is used, which supports the standard corner-point grid format, see [ 24 , 46 ] for more information about corner-point grids. Table 3 lists the discrete error norms, the number of entries in the Jacobian matrix noe , and the number of Newton iterations needed for the simulation run.
It can be seen that all schemes except the TPFA scheme reproduce the exact solution, because the errors are within the range of the nonlinear and linear solver tolerance, whereas the errors of the linear TPFA scheme are approximately five orders of magnitude higher. However, the improved accuracy of the other schemes comes with the cost of a larger face flux stencil, which is the reason why the corresponding Jacobian matrices are denser than the one of the TPFA scheme.
This is due to the fact that the calculation of the coefficient matrix 37 for the MFD and HFV schemes, in general, include all face information, i. Additionally, flux conservation is weakly enforced using the additional equation 11 that is assembled into the global discretization matrix. The MPFA-O or Box scheme cannot be easily applied to this test case because of the non-matching interface between the sub-domains. Nabet have been swapped. More details are available on the Timetable page. Objectives of the conference The finite volume method in its numerous variants is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation.
Organizing Committee E. Audusse Univ. Paris 13 C. Chainais-Hillairet Univ. Lille 1 R.