The two-subdomain DD case in the mixed formulation
In this section, we introduce two equivalent formulations of problem (1.1) using domain decomposition methods: the Dirichlet-to-Neumann formulation and the Robinto- Robin formulation. The Dirichlet-to-Neumann method is based on physical transmission conditions, whereas the Robin-to-Robin method is based on Robin transmission conditions. For simplicity, we present these methods for the two-subdomain case, where N = 2. For example, starting from Figure 1.1, we decompose into two nonoverlapping subdomains 1 and 2 as illustrated in Figure 1.2. We denote the interface by 1,2. As the problems in the subdomains are locally similar in native to the original problem, we will be able to solve all of them using the same solver.
Discrete problem: approximation by the mixed finite element method
Many different numerical methods for the discretization of partial differential equations can be used to approximate their solution. The conservative cell-centered techniques, as the mixed finite element method, mixed hybrid finite element method, or finite volume method are suitable here because they give the conservation of mass which is essential in our application. In this work, we will use the mixed finite element method with the lowest-order Raviart–Thomas–Nédélec space [136, 142]. Therefore, the scalar variable p is approximated in L2 by a constant in each element of the mesh, and the vector function u is approximated in H(div) by functions such that the divergence
is constant on each element and the normal trace is constant over the edges in two dimensions and faces in three dimensions. Having the weak mixed formulation (1.59), we can apply the mixed finite element method, by replacing the spaces Mi and Wi by their finite-dimensional subspaces Mh,i and Wh,i.
The MFE method in one domain with different boundary conditions
This first example focuses on using the MFE method to solve the problem (1.1) with Robin boundary condition −u · n + p on one part of ∂ , where on =]0,1[×]0,1[. We have:
• the diffusion tensor: S = 3 2 2 3 .
• the exact solution: p(x, y) = sin(2πx) sin(2πy).
• f =24π2 sin(2πx) sin(2πy)-16π2 cos(2πx) cos(2πy) the corresponding source term.
• x = 0 the Neumann boundary.
• x = 1 the Robin boundary.
• y = 0 and y = 1 the Dirichlet boundaries.
where the boundary conditions are prescribed by the exact solution.
As explained in the previous sections, the appoximate solution ph of p by the MFE method is constant on each mesh element, see Figure 1.10 for |Th| = 512 triangles.
The MFE method in the DD method: Jacobi iterative solver
Let us first consider the domain decomposition of =]0,1[×]0,1[ into two subdomains 1 =]0,0.5[×]0,1[ and 2 =]0.5,1[×]0,1[. The global numbering of the domain becomes an independent numbering in each subdomain, as shown in Figures 1.14–1.15. The same problem defined above is now solved on each subdomain. In this part, the discrete counterpart of the interface problem (1.46) is solved with the Jacobi algorithm. The DD stopping criteria is when the jump of Robin condition on the interface is less than 1e-10. In Figure 1.16 we show the convergence rate of the pressure error (on the left) and the flux error (on the right). The values of the error are shown in Tables 1.3 and 1.4 again to give more precision. It is clear from Figure 1.17 that the difference between the approximate solution ph in the monodomain case and the approximate solution ph,DD in the DD method is very small. We observe that the DD algorithm converges up to the given tolerance 1e-10 to the discrete monodomain solution (the solution of the discrete counterpart of problem 1.1). We can see in Table 1.5 that this difference is also very small for different mesh refinements.
For all algebraic and domain decomposition solvers fully converged and without the presence of rounding errors, the approximate solution and the monodomain solution theoretically coincide. In this Chapter, the tolerance in the stopping criterion of the DD algorithm is chosen arbitrary equal to 1e-10 which maybe a too much restrictive stopping criterion. In the next chapter, we will develop the theory of a posteriori estimates for the DD method in order to obtain an a posteriori stopping criterion with a lower tolerance, adapted to the problem considered.
Improving k+1 h to obtain the balance with the source term
We will now define an area, called a band, which contains triangles that share an edge or a vertex with i, j . Inspired by , we will construct k+1 h ∈ H(div, ), which is the solution of a local Neumann problem in the band. We use the MFE method to solve this local problem in order to obtain k+1 h . The crucial point is to find Neumann conditions on the boundary of the band that are in equilibrium with the prescribed source term whereas in  the equilibrium is the result of a fixed assumption which is not valid in our case. For simplicity, we first consider the case of the DD where any subdomain touches the boundary ∂ . To explain the idea, we start with the twosubdomain
case 1 and 2: for i = 1,2, we split i into two parts ext i and int i such that ext i ∪ int i = i . Note that 1,2 ⊂ ext 1 ⊂ 1, and also 1,2 ⊂ ext 2 ⊂ 2, see Figure 2.3. The band ext i is made up of triangles that have an edge or a vertex on the interface 1,2. We also denote b i for i = 1,2 and b = 1,2 the intersections of ∂ ext i with ∂ i ∩ ∂ of nonzero (d −1)-dimensional measure.
Table of contents :
1 Domain decomposition for steady diffusion in mixed formulations
1.1 The diffusion equation
1.2 Presentation of some function spaces
1.3 The two-subdomain DD case in the mixed formulation
1.3.1 Multidomain formulation with physical transmission conditions .
1.3.2 Multidomain formulation with Robin transmission conditions
1.4 The case of many subdomains using the Optimized Schwarz Method
1.4.1 Local solver of the Jacobi method
1.5 The local solver in the mixed finite element formulation
1.5.1 Continuous problem: weak mixed formulation
1.5.2 Discrete problem: approximation by the mixed finite element method
1.6 Numerical results
1.6.1 The MFEmethod in one domain with different boundary conditions
1.6.2 The MFE method in the DD method: Jacobi iterative solver .
2 Estimates and stopping criteria in steady diffusion case
2.1 Postprocessing of pk+1 h in the lowest-order Raviart–Thomas case
2.2 Concept of potential and flux reconstructions
2.2.1 Potential reconstruction
2.2.2 Subdomain potential reconstruction
2.2.3 Equilibrated flux reconstruction
2.3 General a posteriori error estimates for ˜ph ∈ H1(Th) and uh ∈ L2( )
2.4 Properties of uk+1 h and pk+1 h in at each iteration of the DD algorithm .
2.5 Potential reconstructions for the Robin DD in the MFE method
2.5.1 Potential reconstruction
2.5.2 Subdomain potential reconstruction
2.6 Flux reconstruction for the Robin DD in the MFE method
2.6.1 Construction of k+1 h ∈ H(div, )
2.6.2 Improving k+1 h to obtain the balance with the source term .
2.7 Numerical results
2.7.1 Example 1 with the Jacobi solver
2.7.2 Example 1 with the GMRES solver
2.7.3 Example 2 with the GMRES solver
3 Estimates and stopping criteria in unsteady diffusion case
3.1 The heat equation
3.2 The global-in-time Optimized Schwarz method using OSWR
3.3 Local solver of the OSWR method for the heat equation
3.4 Local solver of the heat equation in the mixed finite element formulation
3.5 Discretization using MFE in space and an implicit scheme in time
3.6 Concept of potential and flux reconstruction for the heat equation
3.6.1 Potential reconstruction
3.6.2 Subdomain potential reconstruction
3.6.3 Equilibrated flux reconstruction
3.7 General a posteriori error estimate
3.8 Potential and flux reconstructions for the global-in-time DD in the MFE method
3.8.1 Potential reconstruction
3.8.2 Subdomain potential reconstruction
3.8.3 Flux reconstruction
3.9 Numerical results
3.9.1 Model example with the Jacobi solver
3.9.2 Model example with the GMRES solver
3.9.3 Example in an industrial context using conforming time grids .
3.10 Global-in-time domain decomposition using nonconforming time grids .
3.11 A posteriori error estimates for nonconforming time grids
3.12 Numerical results
3.12.1 Example in an industrial context using nonconforming time grids
4 Estimates and stopping criteria in a two-phase flow problem
4.2 Presentation of the problem
4.2.1 Flow between two rock types
4.2.2 Transformation of the equations and weak formulation
4.3 Space-Time Domain Decomposition Methods with Ventcell Transmission
4.4 The cell-centered finite volume scheme
4.4.1 Space-time discretization, notations, and function spaces
4.4.2 A space-time fully discrete scheme based on finite volumes in space and the backward Euler scheme in time
4.4.3 Newton linearization
4.5 Postprocessing and H1- and H(div)-conforming reconstructions
4.5.1 Discontinuous piecewise quadratic ˜ϕ k,n,m h,i and postprocessed saturation uk,n,m h,i
4.5.2 Continuous piecewise quadratic ˆ ϕk,n,m h,i and H1-conforming reconstruction sk,m hτ
4.5.3 Equilibrated flux reconstruction k,m hτ
4.6 A posteriori error estimate
4.7 An a posteriori error estimate distinguishing the space, time, linearization, and the DD errors
4.8 Stopping criteria and optimal balancing of the different error components
4.9 Numerical experiments
4.9.1 The performance of the OSWR method with adaptive stopping criteria
4.9.2 Comparison of Robin- and Ventcell-OSWR algorithm with adaptive stopping criteria
Conclusions and future work
Appendix A Vectorisation in MATLAB