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Problems arising in Earth's mantle convection involve finding the solution to Stokes systems with large viscosity contrasts. These systems contain localized features which, even with adaptive mesh refinement, result in linear systems that can be on the order of 109 or more unknowns. One common approach for preconditioning to the velocity block of these systems is to apply an Algebraic Multigrid (AMG) V‐cycle (as is done in the ASPECT software, for example), however, we find that AMG is lacking robustness with respect to problem size and number of parallel processes. Additionally, we see an increase in iteration counts with refinement when using AMG. In contrast, the Geometric Multigrid (GMG) method, by using information about the geometry of the problem, should offer a more robust option.Here we present a matrix‐free GMG V‐cycle which works on adaptively refined, distributed meshes, and we will compare it against the current AMG preconditioner (Trilinos ML) used in the ASPECT1 software. We will demonstrate the robustness of GMG with respect to problem size and show scaling up to 114,688 cores and 217 billion unknowns. All computations are run using the open‐source, finite element library deal.II.2 [ABSTRACT FROM AUTHOR]

Computations have helped elucidate the dynamics of Earth's mantle for several decades already. The numerical methods that underlie these simulations have greatly evolved within this time span, and today include dynamically changing and adaptively refined meshes, sophisticated and efficient solvers, and parallelization to large clusters of computers. At the same time, many of the methods - discussed in detail in a previous paper in this series - were developed and tested primarily using model problems that lack many of the complexities that are common to the realistic models our community wants to solve today. With several years of experience solving complex and realistic models, we here revisit some of the algorithm designs of the earlier paper and discuss the incorporation of more complex physics. In particular, we re-consider time stepping and mesh refinement algorithms, evaluate approaches to incorporate compressibility, and discuss dealing with strongly varying material coefficients, latent heat, and how to track chemical compositions and heterogeneities. Taken together and implemented in a high-performance, massively parallel code, the techniques discussed in this paper then allow for high resolution, 3-D, compressible, global mantle convection simulations with phase transitions, strongly temperature dependent viscosity and realistic material properties based on mineral physics data. [ABSTRACT FROM AUTHOR]

Melt generation and migration are an important link between surface processes and the thermal and chemical evolution of the Earth's interior. However, their vastly different timescales make it difficult to study mantle convection and melt migration in a unified framework, especially for 3-D global models. And although experiments suggest an increase in melt volume of up to 20 per cent from the depth of melt generation to the surface, previous computations have neglected the individual compressibilities of the solid and the fluid phase. Here, we describe our extension of the finite element mantle convection code ASPECT that adds melt generation and migration. We use the original compressible formulation of the McKenzie equations, augmented by an equation for the conservation of energy. Applying adaptive mesh refinement to this type of problems is particularly advantageous, as the resolution can be increased in areas where melt is present and viscosity gradients are high, whereas a lower resolution is sufficient in regions without melt. Together with a high-performance, massively parallel implementation, this allows for high-resolution, 3-D, compressible, global mantle convection simulations coupled with melt migration. We evaluate the functionality and potential of this method using a series of benchmarks and model setups, compare results of the compressible and incompressible formulation, and show the effectiveness of adaptive mesh refinement when applied to melt migration. Our model of magma dynamics provides a framework for modelling processes on different scales and investigating links between processes occurring in the deep mantle and melt generation and migration. This approach could prove particularly useful applied to modelling the generation of komatiites or other melts originating in greater depths. The implementation is available in the Open Source ASPECT repository. [ABSTRACT FROM AUTHOR]

SUMMARY Numerical simulation of the processes in the Earth's mantle is a key piece in understanding its dynamics, composition, history and interaction with the lithosphere and the Earth's core. However, doing so presents many practical difficulties related to the numerical methods that can accurately represent these processes at relevant scales. This paper presents an overview of the state of the art in algorithms for high-Rayleigh number flows such as those in the Earth's mantle, and discusses their implementation in the Open Source code Aspect (Advanced Solver for Problems in Earth's ConvecTion). Specifically, we show how an interconnected set of methods for adaptive mesh refinement (AMR), higher order spatial and temporal discretizations, advection stabilization and efficient linear solvers can provide high accuracy at a numerical cost unachievable with traditional methods, and how these methods can be designed in a way so that they scale to large numbers of processors on compute clusters. Aspect relies on the numerical software packages deal.II and Trilinos, enabling us to focus on high level code and keeping our implementation compact. We present results from validation tests using widely used benchmarks for our code, as well as scaling results from parallel runs. [ABSTRACT FROM AUTHOR]

Mantle convection models based on geophysical constraints have provided us with a basic understanding of the forces driving and resisting plate motions on Earth. However, existing studies computing the balance of underlying forces are contradicting, and the impact of plate boundary geometry on surface deformation remains unknown. We address these issues by developing global instantaneous 3‐D mantle convection models with a heterogeneous density and viscosity distribution and weak plate boundaries prescribed using different geometries. We find that the plate boundary geometry of the Global Earthquake Model (GEM, Pagani et al., 2018, https://doi.org/10.1177/8755293020931866), featuring open plate boundaries with discrete lithospheric‐depth weak zones in the oceans and distributed crustal faults within continents, achieves the best fit to the observed GPS data with a directional correlation of 95.1% and a global point‐wise velocity residual of 1.87 cm/year. A good fit also requires plate boundaries being 3 to 4 orders of magnitude weaker than the surrounding lithosphere and low asthenospheric viscosities between 5 × 1017 and 5 × 1018 Pa s. Models without asthenospheric and lower mantle heterogeneities retain on average 30% and 70% of the plate speeds, respectively. Our results show that Earth's plate boundaries are not uniform and better described by more discrete plate boundaries within the oceans and distributed faults within continents. Furthermore, they emphasize the impact of plate boundary geometry on the direction and speed of plate motions and reaffirm the importance of slab pull in the uppermost mantle as a major plate driving force. Plain Language Summary: Plate tectonics can explain several geological and geophysical phenomena on Earth and is closely coupled to convection in the underlying mantle. To understand this plate–mantle coupling and quantify the forces contributing to plate motion, we develop high‐resolution three‐dimensional computational models of the Earth's present‐day mantle flow utilizing available geophysical constraints on density distribution and rheology. Additionally, we prescribe weak zones at the location of plate boundaries. We use different plate boundary geometries, forming either open or closed polygons, and we vary how easily the plate boundaries and the asthenosphere directly below the plates can be deformed to determine which model best fits observed plate motions. Our best‐fitting model features open plate boundaries that are weak (∼4 orders of magnitude weaker than the surrounding lithosphere) and traverse the whole plate in the oceans, but are shallower and more distributed within continents. The asthenosphere in these models is even weaker than the plate boundaries. Furthermore, we find that the downward force caused by subducted slabs contributes the most to the observed surface velocities. Our models suggest that plate boundaries are not uniformly weak everywhere and that their geometry has a strong influence on the direction and speed of plate motion. Key Points: We model plate motions in global instantaneous 3‐D mantle convection models with different plate boundary geometriesEarth's plate boundaries are not uniform and better described by discrete shear zones in the oceans and distributed faults within continentsSlab pull within the uppermost mantle (<300 km depth) contributes about 70% of the total plate driving force [ABSTRACT FROM AUTHOR]

Geodynamic simulations increasingly rely on models with a true free surface to investigate questions of dynamic topography, tectonic deformation, gravity perturbations, and global mantle convection. However, implementations of free surface boundary conditions have proven challenging from a standpoint of accuracy, robustness, and stability. In particular, time integration of a free surface tends to suffer from a numerical instability that manifests as sloshing surface motions, also known as the “drunken sailor” instability. This instability severely limits stable timestep sizes to those much smaller than can be used in geodynamic simulations without a free surface. Several schemes have been proposed in the literature to deal with these instabilities. Here we analyze the problem of creeping viscous flow with a free surface and discuss the origin of these instabilities. We demonstrate their cause and how existing stabilization schemes work to damp them out. We also propose a new scheme for removing instabilities from free surface calculations. It does not require modifications to the system matrix, nor additional variables, but is instead an explicit scheme based on nonstandard finite differences. It relies on a single stabilization parameter which may be identified with the smallest relaxation timescale of the free surface. Finally, we present numerical results to show the effectiveness of the new approach and discuss the free surface implementation in the open source, community based mantle convection software ASPECT . [ABSTRACT FROM AUTHOR]