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Summation-by-parts (SBP) finite difference methods are widely used in scientific applications alongside a special treatment of boundary conditions through the simultaneous-approximate-term (SAT) technique which enables the valuable proof of numerical stability. Our work is motivated by multi-scale earthquake cycle simulations described by partial differential equations (PDEs) whose discretizations lead to huge systems of equations and often rely on iterative schemes and parallel implementations to make the nu- merical solutions tractable. In this study, we consider 2D, variable coefficient elliptic PDEs in complex geometries discretized with the SBP-SAT method. The multigrid method is a well-known, efficient solver or preconditioner for traditional numerical discretizations, but they have not been well-developed for SBP-SAT methods on HPC platforms. We propose a custom geometric-multigrid pre- conditioned conjugate-gradient (MGCG) method that applies SBP- preserving interpolations. We then present novel, matrix-free GPU kernels designed specifically for SBP operators whose differences from traditional methods make this task nontrivial but that perform 3× faster than SpMV while requiring only a fraction of memory. The matrix-free GPU implementation of our MGCG method per- forms 5× faster than the SpMV counterpart for the largest problems considered (67 million degrees of freedom). When compared to off- the-shelf solvers in the state-of-the-art libraries PETSc and AmgX, our implementation achieves superior performance in both itera- tions and overall runtime. The method presented in this work offers an attractive solver for simulations using the SBP-SAT method. 

ABSTRACT Numerical modeling of earthquake dynamics and derived insight for seismic hazard relies on credible, reproducible model results. The sequences of earthquakes and aseismic slip (SEAS) initiative has set out to facilitate community code comparisons, and verify and advance the next generation of physics-based earthquake models that reproduce all phases of the seismic cycle. With the goal of advancing SEAS models to robustly incorporate physical and geometrical complexities, here we present code comparison results from two new benchmark problems: BP1-FD considers full elastodynamic effects, and BP3-QD considers dipping fault geometries. Seven and eight modeling groups participated in BP1-FD and BP3-QD, respectively, allowing us to explore these physical ingredients across multiple codes and better understand associated numerical considerations. With new comparison metrics, we find that numerical resolution and computational domain size are critical parameters to obtain matching results. Codes for BP1-FD implement different criteria for switching between quasi-static and dynamic solvers, which require tuning to obtain matching results. In BP3-QD, proper remote boundary conditions consistent with specified rigid body translation are required to obtain matching surface displacements. With these numerical and mathematical issues resolved, we obtain excellent quantitative agreements among codes in earthquake interevent times, event moments, and coseismic slip, with reasonable agreements made in peak slip rates and rupture arrival time. We find that including full inertial effects generates events with larger slip rates and rupture speeds compared to the quasi-dynamic counterpart. For BP3-QD, both dip angle and sense of motion (thrust versus normal faulting) alter ground motion on the hanging and foot walls, and influence event patterns, with some sequences exhibiting similar-size characteristic earthquakes, and others exhibiting different-size events. These findings underscore the importance of considering full elastodynamics and nonvertical dip angles in SEAS models, as both influence short- and long-term earthquake behavior and are relevant to seismic hazard.