Multiscale mortar mixed finite element methods for the Biot system of poroelasticity

TL;DR

This work develops a multiscale mortar mixed finite element method for the Biot poroelastic system on non-matching subdomain grids. By introducing a displacement-pressure mortar Lagrange multiplier and a five-field weakly symmetric formulation, the authors obtain a robust, provably well-posed discretization with rigorous error estimates. The method reduces the global problem to a positive definite interface problem and employs a multiscale stress–flux basis to decouple solver effort from iteration count and time steps, yielding substantial computational savings in heterogeneous settings. Numerical experiments on benchmark and SPE10-like data demonstrate accuracy comparable to fine-scale methods with significantly lower cost and strong scalability potential for parallel computing.

Abstract

We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement-pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish existence, uniqueness, stability, and error estimates for the semidiscrete continuous-in-time formulation under a suitable condition on the richness of the mortar space. We further consider a fully-discrete method based on the backward Euler time discretization and show that the solution of the algebraic system at each time step can be reduced to solving a positive definite interface problem for the composite mortar variable. A multiscale stress-flux basis is constructed, which makes the number of subdomain solves independent of the number of iterations required for the interface problem, as well as the number of time steps. We present numerical experiments verifying the theoretical results and illustrating the multiscale capabilities of the method for a heterogeneous benchmark problem.

Figures (6)

• Figure 1: Example 1, left: physical and numerical parameters; right: coarsest non-matching subdomain grids.
• Figure 2: Example 1, computed solution at the final time step using a linear mortar on non-matching subdomain grids, $h=1/32$, $\Delta t=10^{-3}$ and $c_{0}=1.0$; top: $x$-stress (left), $y$-stress (middle), displacement (right); bottom: rotation (left), velocity (middle), pressure (right).
• Figure 3: Example 2, permeability, porosity, and Young's modulus.
• Figure 4: Example 2, pressure (color) and velocity (arrows); from left to right: fine scale, single linear mortar per interface, two linear mortars per interface, single quadratic mortar per interface (left), two quadratic mortars per interface.
• Figure 5: Example 2, velocity magnitude; from left to right: fine scale, single linear mortar per interface, two linear mortars per interface, single quadratic mortar per interface (left), two quadratic mortars per interface.

Theorems & Definitions (28)

• Remark 2.1
• Remark 3.1
• Lemma 3.1: Pressure mortar inf-sup condition
• proof
• Lemma 3.2: Displacement mortar inf-sup condition
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof