## 1 Introduction

### 1.1 Computational Geodynamics

[2] Over geological time scales, the dynamics of the Earth's mantle and lithosphere can be described by conservation equations applicable to highly viscous, creeping fluids. Fluid motion is largely driven by the heat produced from the Earth's core and from radiogenic heat sources released from the Earth's mantle. Given our present-day understanding of the Earth's rheological and compositional structure with depth, it is apparent that this coupled system is driven by inherently multiscale processes. Considering only the compositional layering, we observe a wide range of relevant length scales. For example, sedimentary and volcanic processes lead to the formation of centimetric to kilometric lithological units, localization of deformation can occur from the millimetric up to the kilometric scale, topographic variations occur on the kilometric scale, tectonic motions involve plates of several thousands of kilometers separated by quasi-discrete plate boundaries, length scale of mantle heterogeneities may be on the order of thousands of kilometers, and processes such as core formation take place at planetary scale.

[3] It is well established in the Earth science community that the use of numerical models to study the long-term geodynamics is a powerful tool to further our understanding of the dynamics of coupled thermomechanical systems. Furthermore, regional (lithospheric) scale numerical models have demonstrated that the inclusion of an upper/lower crust, topography and surface processes can have a profound influence on the dynamics of the lithosphere [*Beaumont et al*., 2001; *Burov et al*., 2001; *Gerya et al*., 2000; *Gorczyk et al*., 2007]. Although the importance of lithosphere-asthenosphere coupling on geodynamics processes is well accepted [*Burov et al*., 2001; *van Hunen et al*., 2000; *Gerya et al*., 2004], many of these modeling studies focused on the deformation within the lithosphere and replaced part of the asthenosphere with various types of boundary conditions (i.e., Winkler) to mimic the effect of mantle flow. Such approximations were necessary to ensure that the essential physics could be resolved using the currently available numerical methods and computational hardware. Specifically, the aforementioned studies that utilized thermomechanical models which employed structured grids, and consequently, simultaneously resolving the entire mantle-lithosphere, the crust, and the topographic variations in a consistently coupled manner, were computationally intractable.

### 1.2 Discretization Techniques

[4] Across the large time scales associated with geologic processes, the underlying rocks (or material lithology) are subject to severe mixing and stirring. To discretize both the viscous flow equations and simultaneously represent and follow the evolution of such large deformations, computational geodynamists frequently advocate the use of a combined mesh-marker approach [*Weinberg and Schmeling*, 1992; *Poliakov and Podladchikov*, 1992; *Zaleski and Julien*, 1992; *Fullsack*, 1995; *Babeyko et al*., 2002; *Gerya and Yuen*, 2003, 2007; *Moresi et al*., 2003, 2007]. In such methods, the flow equations are discretized on the mesh, whilst Lagrangian markers are utilized to discretize the material lithology and history variables.

[5] The use of the classical staggered finite difference (SGFD) scheme [*Harlow and Welch*, 1965] to solve the creeping flow equations (Stokes flow) for lithospheric-scale geodynamic problems has been demonstrated to be both (i) practical, in terms of the computational resources required [e.g., *Gerya and Yuen*, 2003, 2007; *Gerya*, 2010; *Petersen et al*., 2010], and (ii) reliable, with respect to the quality of the numerical solution [*Duretz et al*., 2011]. The practicality of the discretization stems from the restriction of a coordinate-aligned, structured grid and the low-order nature of the stencil which is used to discretize velocity (linear) and pressure (constant). The low-order discretization results in second-order accurate solutions for velocity and pressure in the *L*_{1} norm when the viscosity is a smooth function and a first-order accurate method (in both velocity and pressure, measured in *L*_{1}) when large discontinuous viscosity variations intersect the pressure control volume [*Duretz et al*., 2011]. The latter scenario is of most relevance in both regional and global geodynamic simulations involving mobile lithospheric plates [*Tackley*, 2000; *Stadler et al*., 2010; *Crameri et al*., 2012a].

[6] The simplicity of the staggered grid discretization is also arguably its biggest weakness. The ability to deform the mesh, for example, to conform to an evolving topography is not permitted in the classical staggered grid formulations. This shortcoming has been addressed by a number of different methods which permit zero normal stress boundary conditions (or approximate boundary conditions) to be applied on the upper surface [*Harlow and Welch*, 1965; *Matsumoto and Tomoda*, 1983; *Leveque and Li*, 1997; *Fedkiw et al*., 1999; *Chern and Shur*, 2007; *Suckale et al*., 2010; *Crameri et al*., 2012b].

### 1.3 Toward an Adaptive Finite Difference Stencil for Stokes Flow

[7] To incorporate local variations in spatial resolution throughout the computational domain, for example, if we wished to resolve both the mantle and the upper/lower crust, several different types of meshes can be considered. For example, a set of quadrilaterals (hexahedra in 3-D) with an unstructured element connectivity or a set of unstructured triangular (tetrahedra in 3-D) could be employed. Having such versatility within the choice of mesh enables one to (i) easily track the free surface and, if utilizing unstructured meshes, (ii) locally decrease the element size *h* in regions of interest. Another alternative mesh permitting local mesh refinement is a block-structured mesh which can be conveniently described via a quadtree (octree in 3-D). See Figure 1 for examples of unstructured triangular and block-structured meshes with local refinement.

[8] One of the major arguments to utilize the finite element (FE) method for adaptive mesh calculations is that this discretization naturally permits a large degree of geometric flexibility in defining the mesh. However, the stencil associated with a stable finite element discretization, *Q*_{2}*P*_{1} (for example) possess far more degrees of freedom than the SGFD stencil. For this reason, much research has focused on generalizing the SGFD methodology to facilitate the use of unstructured adaptive meshes.

[9] Fully unstructured generalizations of the staggered discretization are indeed possible [*Hirt et al*., 1974; *Rhie and Chow*, 1983; *Reggio and Camarero*, 1986; *Rodi et al*., 1989] and fall under the class of finite volume methods. Whilst such methods alleviate the limitations of having a coordinate-aligned structured grid, the resulting stencils are larger than the classical SGFD stencil—thus increasing the storage requirements.

[10] To circumvent the cost and programming complexity associated with generalized staggered grid formulations, numerous velocity-pressure arrangements were developed. The most attractive being a co-located (cell-centered or vertex-based) discretization in which both velocity components were defined at the same point in space. Co-located arrangements however suffer from nonphysical oscillations in the pressure field [*Patankar*, 1980]. Extensive comparisons of results obtained with staggered and non-staggered grid arrangements have been conducted [*Perič et al*., 1988; *Miller and Schmidt*, 1988; *Shih et al*., 1989; *Armfield*, 1991; *Aksoy and Chen*, 1992; *Melaaen*, 1992a, 1992b; *Choi et al*., 1994a, 1994b]. Remedies to circumvent the spurious pressure oscillations associated with co-located discretizations have been developed [*Rhie and Chow*, 1983; *Miller and Schmidt*, 1988; *Majumdar*, 1988; *Miller and Schmidt*, 1988; *Papageorgakopoulos et al*., 2000].

[11] We note that all comparisons conducted between different finite difference stencils and the stabilization techniques developed for co-located formulations focus on constant viscosity, Navier-Stokes equations. The time dependence introduced in the Navier-Stokes equations permits certain time-splitting discretizations (or decoupling) to be employed which further relax the need for properly coupled velocity-pressure discretizations. In geodynamic applications, we specifically require methods which are robust for fluids possessing highly spatially variable viscosity and which are in the steady state Stokes regime (non-inertial). In our experience, the most robust and reliable finite difference discretizations for variable viscosity Stokes flow are the fully staggered formulations [*Gerya*, 2010].

[12] An alternative approach to using an unstructured staggered grid was the fully adaptive, block-structured orthogonal staggered grid finite difference method of *Albers* [2000] which was developed specifically for studying mantle convection, in which the flow problem possessed smooth variations in viscosity. The methodology developed in this work is simple and maintains a compact stencil. At the transition between different grid resolutions, such meshes produce “hanging” nodes—i.e., the nodes which are only contained within the region of finer resolution. At these hanging nodes, the stencil adopted in *Albers* [2000] employed direct velocity and pressure interpolation. In general, this method cannot be applicable for modeling lithospheric problems with sharply variable viscosity since direct velocity interpolation does not ensure conservation of stresses across resolution boundaries with large viscosity changes. For this common geodynamic modeling situation, a stress-conservative discretization is required [*Gerya and Yuen*, 2003, 2007].

### 1.4 Present Work

[13] To address the need of a low-order, robust, and practical adaptive discretization for computational geodynamics, we extend the methodology presented in *Albers* [2000]. To this end, we develop a fully adaptive, block-structured orthogonal staggered grid finite difference stencil. The crucial component of the method developed here is the use of stress-conservative finite difference schemes across split-cell faces. This proves to be of fundamental importance to eliminate spurious pressure oscillations across cells with split faces. Most importantly, we examine the stability and order of accuracy of the stress-conservative adaptive staggered grid finite difference stencil for a range of viscosity structures.

[14] The outline of the paper is as follows. In section 2, we describe the governing equations of creeping flow and define how to construct staggered grid stencils on quadtree-based adaptive meshes. In section 3, we demonstrate the stability of the adaptive staggered grid discretization and demonstrate the order of accuracy of the method using several analytic solutions for variable viscosity Stokes flow in section 3.2. Furthermore, we highlight the computational advantage obtained using adaptive grids compared to the classical, nonadaptive staggered grids in section 4. Practical examples of lithospheric and planetary scales models are presented in section 5. Lastly, insections 6 and 7, we summarize the adaptive staggered grid formulation and indicate the future directions and possibilities of this methodology.