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Implicit-Explicit Scheme with Multiscale Vanka Two-Grid Solver for Heterogeneous Unsaturated Poroelasticity

Maria Vasilyeva, Ben S. Southworth, Yunhui He, Min Wang

Abstract

We consider a coupled nonlinear system of equations that describe unsaturated flow in heterogeneous poroelastic media. For the numerical solution, we use a finite element approximation in space and present an efficient multiscale two-grid solver for solving the coupled system of equations. The proposed two-grid solver contains two main parts: (i) accurate coarse grid approximation based on local spectral spaces and (ii) coupled smoothing iterations based on an overlapping multiscale Vanka method. A Vanka smoother and local spectral coarse grids come with significant computational cost in the setup phase. To avoid constructing a new solver for each time step and/or nonlinear iteration, we utilize an implicit-explicit integration scheme in time, where we partition the nonlinear operator as a sum of linear and nonlinear parts. In particular, we construct an implicit linear approximation of the stiff components that remains fixed across all time, while treating the remaining nonlinear residual explicitly. This allows us to construct a robust two-grid solver offline and utilize it for fast and efficient online time integration. A linear stability analysis of the proposed novel coupled scheme is presented based on the representation of the system as a two-step scheme. We show that the careful decomposition of linear and nonlinear parts guarantees a linearly stable scheme. A numerical study is presented for a two-dimensional nonlinear coupled test problem of unsaturated flow in heterogeneous poroelastic media. We demonstrate the robustness of the two-grid solver, particularly the efficacy of block smoothing compared with simple pointwise smoothing, and illustrate the accuracy and stability of implicit-explicit time integration.

Implicit-Explicit Scheme with Multiscale Vanka Two-Grid Solver for Heterogeneous Unsaturated Poroelasticity

Abstract

We consider a coupled nonlinear system of equations that describe unsaturated flow in heterogeneous poroelastic media. For the numerical solution, we use a finite element approximation in space and present an efficient multiscale two-grid solver for solving the coupled system of equations. The proposed two-grid solver contains two main parts: (i) accurate coarse grid approximation based on local spectral spaces and (ii) coupled smoothing iterations based on an overlapping multiscale Vanka method. A Vanka smoother and local spectral coarse grids come with significant computational cost in the setup phase. To avoid constructing a new solver for each time step and/or nonlinear iteration, we utilize an implicit-explicit integration scheme in time, where we partition the nonlinear operator as a sum of linear and nonlinear parts. In particular, we construct an implicit linear approximation of the stiff components that remains fixed across all time, while treating the remaining nonlinear residual explicitly. This allows us to construct a robust two-grid solver offline and utilize it for fast and efficient online time integration. A linear stability analysis of the proposed novel coupled scheme is presented based on the representation of the system as a two-step scheme. We show that the careful decomposition of linear and nonlinear parts guarantees a linearly stable scheme. A numerical study is presented for a two-dimensional nonlinear coupled test problem of unsaturated flow in heterogeneous poroelastic media. We demonstrate the robustness of the two-grid solver, particularly the efficacy of block smoothing compared with simple pointwise smoothing, and illustrate the accuracy and stability of implicit-explicit time integration.

Paper Structure

This paper contains 22 sections, 1 theorem, 82 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Unsaturated flow
  4. Mechanics
  5. Problem setting and assumptions
  6. Spatial discretization and semi-implicit time integration
  7. Implicit-explicit scheme for time approximation
  8. Additive scheme
  9. Stability analysis
  10. Choice of the additive representation and system of linear equation
  11. Two-grid solver
  12. Coarse grid solver
  13. Smoothing iterations
  14. Numerical results
  15. Test 1
  16. ...and 7 more sections

Key Result

Theorem 1

Let nonlinear discretization operators be linearized about some valid state, and assume all linearized discretization operators and corresponding linear approximations, e.g. $A_h^{(nl)}$ and $A_h^{n, (lin)}$, are symmetric. Suppose with $0<\varrho<1$. Then the solution of the discrete problem eq:imex1 is stable.

Figures (13)

  • Figure 1: Fine grid (blue color), coarse grid (black color), local domain $\omega_l$, coarse cell $K_i$ and oversampled coarse cell $K_i^o$ for $o=2$ with local fine grid resolution.
  • Figure 2: Coarse grid and groups of subdomains $J^k$ ($k=1,2,3,4$). Left panel: $\iota_i = \omega_i$. Right panel: $\iota_i = K_i^o$.
  • Figure 3: Test 1. Heterogeneous coefficients $k_s(x)$[m$^2$] and $E_d(x)$[Pa], pressure [Pa] and displacements $x$ and $y$ components [cm] at final time (from left to right).
  • Figure 4: Number of nonlinear iterations for implicit scheme with Picard iterations. Shown in legend total number of nonlinear iterations ($N^{tot}_{nl}$).
  • Figure 5: Number of time steps ($N_t$) vs relative error for pressure $e_p$ (left) and displacements $e_u$ (right) at final time.
  • ...and 8 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Remark 1