diff --git a/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md b/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md new file mode 100644 index 0000000..e01c700 --- /dev/null +++ b/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md @@ -0,0 +1,358 @@ +--- +title: Benchmarking Stokes Flow in Annulus and Spherical-Shell Geometries +description: >- + A suite of annulus and spherical-shell Stokes benchmarks reproduced in + Underworld3. What each benchmark is designed to test, what numerical + behaviour to expect from it, and how to read the convergence results — + including where reduced rates are the correct answer rather than a defect. +date: 2026-08-13 +authors: + - name: Thyagarajulu Gollapalli + orcid: 0000-0001-9394-4104 + affiliations: + - Monash University +license: CC-BY-4.0 +bibliography: + - references.bib +banner: figures/banner.jpg +keywords: + - Underworld Code + - benchmarks + - development +exports: + - format: typst + logo: ../../static/uwtn-logo.png + series: "Underworld Technical Notes" + origin_url: https://www.underworldcode.org/annulus-and-spherical-stokes-benchmarks/ + template: ../../templates/pdf + output: annulus-and-spherical-stokes-benchmarks.pdf + article_id: UWTN 2026-012 + article_version: 1.0.0 + software_version: underworld3 0.0.0 +--- +
Photo by the blowup / Unsplash
+ +This post introduces a suite of annulus and spherical-shell Stokes benchmarks +reproduced using Underworld3 [@Moresi_2025]. These benchmark problems have +previously been implemented in several numerical codes, but here we bring them +together within a single framework to highlight both the strengths and +practical challenges of curved-domain finite-element modelling. The post is +aimed at researchers working on geodynamics numerical modelling. Rather than +focusing on heavy mathematical derivations or low-level implementation details, +the goal is to provide an intuitive and practical guide to what each benchmark +is designed to test, what numerical behaviour should be expected, and how to +interpret the results. By the end, readers should have a clear understanding of +the purpose of these benchmarks and the key ideas behind verifying Stokes flow +in curved geometries. + +## What Is Benchmarking and Why Is It Important? + +Benchmarking is the process of testing a numerical method or software +implementation against problems with known analytical solutions or +well-established reference results. In computational geodynamics, benchmarks +are essential because they help verify that a code correctly solves the +governing equations before it is applied to complex Earth-science problems +where the true solution is unknown. A good benchmark does more than produce a +visually reasonable result; it tests numerical convergence behaviour, +boundary-condition implementation, mesh geometry, and solver robustness under +controlled conditions. Curved-domain benchmarks such as annulus and +spherical-shell Stokes problems are particularly important because they expose +numerical challenges that do not appear in simple Cartesian geometries. +Successfully reproducing benchmark results therefore builds confidence that a +numerical framework can model real-world problems with reliable and physically +meaningful approximations. + +## What Are the Stokes Equations? + +The Stokes equations describe the slow, viscous flow of fluids in situations +where inertial forces are negligible compared with viscous forces. In +geodynamics, this approximation is widely used because rocks in the Earth's +mantle deform extremely slowly over geological timescales and behave like +highly viscous fluids. The incompressible Stokes equations are written as + +$$ +\begin{aligned} +-\nabla \cdot \left(2 \eta \dot{\varepsilon}(\mathbf{u})\right) ++ \nabla p &= \rho \mathbf{g}, \\ +\nabla \cdot \mathbf{u} &= 0 . +\end{aligned} +$$ + +Here $\mathbf{u}$ is velocity, $p$ is pressure, $\eta$ is viscosity, $\rho$ is +density, and $\mathbf{g}$ is gravity. The first equation represents +conservation of momentum, balancing viscous stresses, pressure gradients, and +body forces. The second equation enforces mass conservation through +incompressibility. Solving these equations allows us to model mantle +convection, lithospheric deformation, subduction, and many other large-scale +Earth processes. Although the equations appear compact, solving them accurately +in curved geometries with complex boundary conditions and variable material +properties is computationally challenging, which is why benchmark problems are +important. + +## Why Do We Need Curved Geometries? + +Curved geometries are important in geodynamics because the Earth itself is +curved. Many large-scale Earth processes, such as mantle convection, +subduction, plume dynamics, and lithospheric deformation, occur within +spherical or shell-like domains rather than simple rectangular boxes. While +Cartesian geometries are useful for developing intuition and testing numerical +methods, they cannot fully represent radial gravity, curved boundaries, or +global-scale flow patterns. Curved-domain models also introduce additional +numerical challenges, including geometric approximation errors, coordinate +transformations, and the accurate implementation of free-slip or zero-slip +boundary conditions on non-planar surfaces. As a result, annulus and +spherical-shell benchmarks provide a more realistic and demanding test of +Stokes solvers. + +## Benchmark Suite + +In this work, we reproduce four widely used Stokes benchmark suites in curved +geometries using Underworld3 [@Moresi_2025]. The first is the +Thieulot–Puckett annulus benchmark, which provides a smooth analytical solution +in an annulus and is mainly used to test optimal finite-element convergence +behaviour [@ThieulotPuckett_2018]. The second is the Kramer annulus benchmark, +which extends the problem to include both smooth volumetric forcing and +singular delta-function forcing on an internal interface, together with +free-slip and zero-slip boundary conditions [@Kramer_2021]. We then consider +the spherical-shell counterparts of these problems: the Thieulot spherical +benchmark, which tests smooth Stokes flow in spherical geometry with both +constant and radially varying viscosity [@Thieulot_2017], and the Kramer +spherical benchmark, which again introduces internal interface forcing and +reduced solution regularity [@Kramer_2021]. Together, these four benchmark +suites test curved geometries, pressure treatment, mesh approximation, +boundary-condition implementation, smooth and singular forcing, and convergence +behaviour in both two- and three-dimensional Stokes flow. + +```{figure} figures/benchmark-fields.jpg +:alt: Four panels of analytical benchmark fields, showing density distributions for the Thieulot-Puckett annulus, the Kramer annulus, the Thieulot spherical shell and the Kramer spherical shell. + +Analytical benchmark fields used in the annulus and spherical-shell Stokes +benchmark suite. The panels collect the Thieulot–Puckett annulus, Kramer +annulus, Thieulot spherical-shell, and Kramer spherical-shell cases. +``` + +## What Do We Measure? Error Quantification + +The benchmark comparisons use $L_2$-norm errors because they give a single +quantitative measure of the difference between the numerical solution and the +analytical solution over the whole domain. For a computed field $q_h$ and +analytical field $q^{*}$, the absolute volume error is + +$$ +E_{L_2}(q) += +\left( +\int_{\Omega} |q_h-q^*|^2\,\mathrm{d}\Omega +\right)^{1/2}. +$$ + +The corresponding relative volume error is + +$$ +E_{L_2}^{\mathrm{rel}}(q) += +\frac{E_{L_2}(q)}{\|q^*\|_{L_2}} += +\left( +\frac{\int_{\Omega} |q_h-q^*|^2\,\mathrm{d}\Omega} + {\int_{\Omega} |q^*|^2\,\mathrm{d}\Omega} +\right)^{1/2}. +$$ + +The relative form is useful when comparing velocity and pressure errors across +different benchmark cases because it normalises the error by the size of the +analytical solution. For pressure, the numerical and analytical fields are +first compared in the same pressure gauge, since incompressible Stokes pressure +is determined only up to an additive constant. + +Boundary pressure errors are measured separately on the inner and outer +surfaces. For a boundary $\Gamma \in \{\Gamma_{\mathrm{inner}}, +\Gamma_{\mathrm{outer}}\}$, the absolute pressure-trace error is + +$$ +E_{L_2,\Gamma}(p) += +\left( +\int_{\Gamma} |p_h-p^*|^2\,\mathrm{d}\Gamma +\right)^{1/2}. +$$ + +Where the analytical boundary pressure has a nonzero $L_2$ norm, the relative +boundary pressure error is + +$$ +E_{L_2,\Gamma}^{\mathrm{rel}}(p) += +\left( +\frac{\int_{\Gamma} |p_h-p^*|^2\,\mathrm{d}\Gamma} + {\int_{\Gamma} |p^*|^2\,\mathrm{d}\Gamma} +\right)^{1/2}. +$$ + +Convergence is measured by comparing errors across successively refined meshes: + +$$ +\mathrm{rate} += +\frac{ +\log\left(E_{h_1}/E_{h_2}\right) +}{ +\log\left(h_1/h_2\right) +}. +$$ + +Here $h$ is the characteristic cell size. If the mesh is uniformly refined so +that $h_2 = h_1/2$, this becomes + +$$ +\mathrm{rate} += +\log_2\left(\frac{E_{h_1}}{E_{h_2}}\right). +$$ + +The expected behaviour is simple: as $h$ decreases, the error should decrease. +On a log-log convergence plot, a method with error proportional to $C h^{r}$ +appears approximately as a straight line with slope $r$. For smooth Stokes +solutions, stable mixed finite-element pairs have well-defined optimal +convergence expectations; for example, Taylor–Hood $P_2 \times P_1$ commonly +gives third-order velocity and second-order pressure convergence in the volume +$L_2$ norm when the geometry and solution are sufficiently smooth +[@Boffi_2013]. Singular forcing cases are different: the solution is less +regular near the internal interface, so reduced convergence rates are expected +even when the solver is implemented correctly. + +## What Do We Observe? Results of Benchmarks in Underworld3 + +### Volumetric Convergence + +The benchmark results show that Underworld3 reproduces the expected convergence +behaviour for both annulus and spherical-shell Stokes problems. For smooth +analytical solutions, such as the Thieulot annulus and spherical benchmarks, +the Taylor–Hood $P_2 \times P_1$ discretisation achieves close to the +theoretically expected convergence rates, with approximately third-order +velocity convergence and second-order pressure convergence in the volume $L_2$ +norm. Higher-order element pairs further improve accuracy, while lower-order +discretisations show the expected reduction in convergence order. + +For the Kramer benchmarks with smooth forcing, the velocity convergence is +closer to second order because the curved geometry is represented using linear +meshes, consistent with the original benchmark studies [@Kramer_2021]. In the +delta-function forcing cases, the convergence rates reduce significantly +because the internal singular interface lowers the regularity of the analytical +solution. Overall, the volumetric results demonstrate that Underworld3 captures +both optimal convergence behaviour for smooth problems and the expected +degradation in accuracy for singular forcing cases. + +```{figure} figures/figure_5_thieulot_annulus_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Thieulot-Puckett annulus benchmark. + +Velocity and pressure convergence for the Thieulot–Puckett annulus benchmark. +``` + +```{figure} figures/figure_3_kramer_annulus_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Kramer annulus benchmark. + +Velocity and pressure convergence for the Kramer annulus benchmark. +``` + +```{figure} figures/figures_4_5_thieulot_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Thieulot spherical-shell benchmark. + +Velocity and pressure convergence for the Thieulot spherical-shell benchmark. +``` + +```{figure} figures/figure_4_kramer_spherical_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Kramer spherical-shell benchmark. + +Velocity and pressure convergence for the Kramer spherical-shell benchmark. +``` + +### Boundary Pressure Convergence + +Boundary diagnostics provide a more local and often more sensitive measure of +solver accuracy in curved geometries. The volume pressure norm measures an +error over the full domain, but boundary pressure errors isolate the pressure +trace on the surfaces where boundary conditions, radial normal stresses, and +traction-related diagnostics are evaluated. This is useful because geometric +approximation errors and boundary quadrature errors can be more visible on +curved inner and outer boundaries than in a volume-averaged metric. + +For the Thieulot–Puckett annulus benchmark, the $P_2 \times P_1$ boundary +pressure errors decrease close to the expected $O(h^2)$ trend. The inner and +outer boundaries do not have identical error constants, which is acceptable: +the two curves have different radii, different geometric representation errors, +and different boundary integration paths. + +```{figure} figures/figure_p2p1_boundary_pressure_convergence.jpg +:alt: Log-log plot of absolute boundary pressure error against cell size for the Thieulot-Puckett annulus benchmark, with separate curves for the inner and outer boundaries. + +Absolute boundary pressure error convergence for the Thieulot–Puckett annulus +benchmark using the $P_2 \times P_1$ discretisation. Solid curves denote the +inner boundary and dashed curves denote the outer boundary. +``` + +For the Kramer annulus benchmark, the boundary pressure errors converge close +to second order for the smooth cases. The delta-function cases also show +stronger boundary pressure convergence than the corresponding volume pressure +norm because the reduced regularity is localized at the internal interface +rather than on the annulus boundaries. + +```{figure} figures/figure_boundary_pressure_convergence.jpg +:alt: Log-log plot of relative boundary pressure error against cell size for the Kramer annulus benchmark, with separate curves for the inner and outer boundaries. + +Relative boundary pressure error convergence for the Kramer annulus benchmark. +Solid curves denote the inner boundary and dashed curves denote the outer +boundary. +``` + +The Thieulot spherical-shell benchmark gives the corresponding smooth +three-dimensional boundary test. The boundary-pressure panel shows that the +absolute pressure-trace errors for both $m = -1$ and $m = 3$ decrease +systematically with refinement. These errors are shown together with radial +normal-stress errors because the stress diagnostic combines pressure and +velocity-gradient errors at the same curved spherical boundaries. + +```{figure} figures/boundary_metric_convergence.jpg +:alt: Two panels of log-log convergence plots for the Thieulot spherical-shell benchmark: boundary radial normal-stress error on the left and absolute boundary pressure error on the right. + +Boundary radial normal-stress and absolute boundary pressure error convergence +for the Thieulot spherical-shell benchmark. The right panel shows +$E_{L_2,\Gamma}(p)$ on the inner and outer spherical boundaries. +``` + +For the Kramer spherical-shell benchmark, the current boundary figure reports +boundary velocity and radial normal-stress convergence for the free-slip +delta-function case. A separate boundary pressure-trace convergence plot is not +part of the current spherical Kramer article outputs. The radial normal-stress +diagnostic is included here because it is pressure-sensitive: $\sigma_{rr}$ +contains the pressure contribution and therefore tests pressure recovery +together with velocity-gradient recovery on the spherical boundaries. + +```{figure} figures/figure_7_kramer_boundary_convergence.jpg +:alt: Log-log convergence plots of boundary velocity error and radial normal-stress error against cell size for the Kramer spherical-shell benchmark. + +Boundary velocity and radial normal-stress convergence for the Kramer +spherical-shell benchmark. This is the available pressure-sensitive boundary +diagnostic for the spherical Kramer case. +``` + +Overall, the boundary diagnostics complement the volume $L_2$ pressure errors. +Smooth benchmark cases recover approximately second-order pressure-trace +convergence for the $P_2 \times P_1$ pair, while singular forcing primarily +affects the volume pressure norm through reduced regularity at the internal +interface. Persistent differences between inner- and outer-boundary error +constants should therefore be interpreted as geometric and boundary-evaluation +effects rather than as a failure of pressure normalisation. + +The full details are in the article PDFs, which can be viewed directly from the +[benchmark figures and articles directory on GitHub](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/tree/main/docs/benchmarks_figures_and_articles): + +- [Thieulot–Puckett annulus benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/annulus/thieulot/thieulot_annulus_benchmark_article.pdf) +- [Kramer annulus benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/annulus/kramer/kramer_annulus_benchmark_article.pdf) +- [Thieulot spherical-shell benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/spherical/thieulot/thieulot_spherical_benchmark_article.pdf) +- [Kramer spherical-shell benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/spherical/kramer/kramer_spherical_benchmark_article.pdf) + +Those articles contain the analytical fields, discretisation choices, +convergence tables, boundary diagnostics, and full reference details. + +
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diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/banner.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/banner.jpg new file mode 100644 index 0000000..5a24e3e Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/banner.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/benchmark-fields.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/benchmark-fields.jpg new file mode 100644 index 0000000..c3e9fba Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/benchmark-fields.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/boundary_metric_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/boundary_metric_convergence.jpg new file mode 100644 index 0000000..bace5f1 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/boundary_metric_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_3_kramer_annulus_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_3_kramer_annulus_convergence.jpg new file mode 100644 index 0000000..28d228b Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_3_kramer_annulus_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_4_kramer_spherical_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_4_kramer_spherical_convergence.jpg new file mode 100644 index 0000000..2afaaa7 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_4_kramer_spherical_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_5_thieulot_annulus_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_5_thieulot_annulus_convergence.jpg new file mode 100644 index 0000000..4c26356 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_5_thieulot_annulus_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_7_kramer_boundary_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_7_kramer_boundary_convergence.jpg new file mode 100644 index 0000000..b471b49 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_7_kramer_boundary_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_boundary_pressure_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_boundary_pressure_convergence.jpg new file mode 100644 index 0000000..b18eb04 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_boundary_pressure_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_p2p1_boundary_pressure_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_p2p1_boundary_pressure_convergence.jpg new file mode 100644 index 0000000..4994d34 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figure_p2p1_boundary_pressure_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/figures/figures_4_5_thieulot_convergence.jpg b/articles/annulus-and-spherical-stokes-benchmarks/figures/figures_4_5_thieulot_convergence.jpg new file mode 100644 index 0000000..62bcc84 Binary files /dev/null and b/articles/annulus-and-spherical-stokes-benchmarks/figures/figures_4_5_thieulot_convergence.jpg differ diff --git a/articles/annulus-and-spherical-stokes-benchmarks/metadata.yml b/articles/annulus-and-spherical-stokes-benchmarks/metadata.yml new file mode 100644 index 0000000..e65ab25 --- /dev/null +++ b/articles/annulus-and-spherical-stokes-benchmarks/metadata.yml @@ -0,0 +1,36 @@ +# Validated in CI against schemas/article-metadata.schema.json. +# `pixi run validate` checks this and the cross-file invariants a schema cannot +# express -- that the article file is named .md, that canonical_path +# matches the slug, and that no legacy DOI is ever paired with a new registrant. +id: UWTN 2026-012 +slug: annulus-and-spherical-stokes-benchmarks +title: Benchmarking Stokes Flow in Annulus and Spherical-Shell Geometries +article_type: technical-note +status: review +authors: + - name: Thyagarajulu Gollapalli + orcid: 0000-0001-9394-4104 + affiliation: Monash University +publication_date: null +version: 1.0.0 +# The deposit writes archive_doi and repository_record_id when the note is +# published; leave them out until then. `doi` and `doi_registrant` were here +# once and are not fields the schema knows -- every note made from this +# template failed `pixi run validate` on all three of them. +license: CC-BY-4.0 +canonical_path: /annulus-and-spherical-stokes-benchmarks/ +legacy_paths: [] +# Facets, from vocabulary.yml. Both keys must be present even when empty: a +# note with no subject is normal -- many are purely about method. +subjects: + - mantle-convection + - planetary +methods: + - benchmarks-validation + - solvers + - finite-elements +ghost_tags: + - Underworld Code +figures: 9 +banner_credit: "Photo by the blowup / Unsplash" +source: native diff --git a/articles/annulus-and-spherical-stokes-benchmarks/references.bib b/articles/annulus-and-spherical-stokes-benchmarks/references.bib new file mode 100644 index 0000000..54979b8 --- /dev/null +++ b/articles/annulus-and-spherical-stokes-benchmarks/references.bib @@ -0,0 +1,70 @@ +% Pinned rather than resolved from doi.org at build time. Letting MyST fetch +% citations over the network makes the build depend on doi.org answering, which +% failed on CI for UWTN 2026-011 and would have published a broken citation. + +@article{Moresi_2025, + title = {Underworld3: Mathematically Self-Describing Modelling in Python for Desktop, HPC and Cloud}, + volume = {10}, + ISSN = {2475-9066}, + url = {http://dx.doi.org/10.21105/joss.07831}, + DOI = {10.21105/joss.07831}, + number = {112}, + journal = {Journal of Open Source Software}, + publisher = {The Open Journal}, + author = {Moresi, Louis and Mansour, John and Giordani, Julian and Knepley, Matt and Knight, Ben and Graciosa, Juan Carlos and Gollapalli, Thyagarajulu and Lu, Neng and Beucher, Romain}, + year = {2025}, + month = aug, + pages = {7831} +} + +% No DOI: a preprint submitted to Computers & Geosciences, cited in the original +% post by its author-hosted PDF. The URL is the only persistent handle there is. +@unpublished{ThieulotPuckett_2018, + title = {Incompressible Stokes flow in an annulus: an analytical solution and numerical benchmark}, + author = {Thieulot, Cedric and Puckett, Elbridge Gerry}, + year = {2018}, + note = {Preprint submitted to Computers \& Geosciences}, + url = {https://www.math.ucdavis.edu/~egp/PUBLICATIONS/JOURNAL_ARTICLES/SUBMITTED/CAPT-EGP-2018.pdf} +} + +@article{Kramer_2021, + title = {Analytical solutions for mantle flow in cylindrical and spherical shells}, + volume = {14}, + ISSN = {1991-9603}, + url = {http://dx.doi.org/10.5194/gmd-14-1899-2021}, + DOI = {10.5194/gmd-14-1899-2021}, + number = {4}, + journal = {Geoscientific Model Development}, + publisher = {Copernicus GmbH}, + author = {Kramer, Stephan C. and Davies, D. Rhodri and Wilson, Cian R.}, + year = {2021}, + month = apr, + pages = {1899--1919} +} + +@article{Thieulot_2017, + title = {Analytical solution for viscous incompressible Stokes flow in a spherical shell}, + volume = {8}, + ISSN = {1869-9529}, + url = {http://dx.doi.org/10.5194/se-8-1181-2017}, + DOI = {10.5194/se-8-1181-2017}, + number = {6}, + journal = {Solid Earth}, + publisher = {Copernicus GmbH}, + author = {Thieulot, Cedric}, + year = {2017}, + month = nov, + pages = {1181--1191} +} + +@book{Boffi_2013, + title = {Mixed Finite Element Methods and Applications}, + ISBN = {9783642365195}, + ISSN = {0179-3632}, + url = {http://dx.doi.org/10.1007/978-3-642-36519-5}, + DOI = {10.1007/978-3-642-36519-5}, + series = {Springer Series in Computational Mathematics}, + publisher = {Springer Berlin Heidelberg}, + author = {Boffi, Daniele and Brezzi, Franco and Fortin, Michel}, + year = {2013} +} diff --git a/classification.yml b/classification.yml index 3ad124a..a658a5f 100644 --- a/classification.yml +++ b/classification.yml @@ -312,3 +312,8 @@ moving-the-mesh-without-remaking-it: article_type: technical-note subjects: [mantle-convection, earthquakes-faults] methods: [meshing, solvers, parallel-hpc] + +annulus-and-spherical-stokes-benchmarks: + article_type: technical-note + subjects: [mantle-convection, planetary] + methods: [benchmarks-validation, solvers, finite-elements]