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In this paper, we consider phase-field-based fracture propagation in elastic media. The main purpose is the development of a robust and efficient numerical scheme. To enforce crack irreversibility as a constraint, we use a primal-dual active set strategy, which can be identified as a semi-smooth Newton's method. The active set iteration is merged with the Newton iteration for solving the fully-coupled nonlinear partial differential equation discretized using finite elements, resulting in a single, rapidly converging nonlinear scheme. It is well known that phase-field models require fine meshes to accurately capture the propagation dynamics of the crack. Because traditional estimators based on adaptive mesh refinement schemes are not appropriate, we develop a predictorcorrector scheme for local mesh adaptivity to reduce the computational cost. This method is both robust and efficient and allows us to treat temporal and spatial refinements and to study the influence of model regularization parameters. Finally, our proposed approach is substantiated with different numerical tests for crack propagation in elastic media and pressurized fracture propagation in homogeneous and heterogeneous media.

Abstract. We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a course grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar space achieves approximation comparable to the fine scale on its coarse grid by using higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale finite element and variational multiscale methods. We derive a priori error estimates and show, with appropriate choice of the mortar space, optimal order convergence and some superconvergence on the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate subdomain and mortar grids. Numerical experiments are presented in confirmation of the theory.

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We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart-Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the L 2-and H −s-norms (and superconvergence is obtained between the L 2-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If h denotes the maximal mesh spacing, then the optimal rate is O(h). The superconvergence rate O(h 2) is obtained for the scalar unknown and rate O(h 3/2) for its gradient and flux in certain discrete norms; moreover, the full O(h 2) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.