Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Algorithm 839: FIAT, a new paradigm for computing finite element basis functions
ACM Transactions on Mathematical Software (TOMS)
Mass preserving finite element implementations of the level set method
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
A compiler for variational forms
ACM Transactions on Mathematical Software (TOMS)
Efficient compilation of a class of variational forms
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Automated Code Generation for Discontinuous Galerkin Methods
SIAM Journal on Scientific Computing
DOLFIN: Automated finite element computing
ACM Transactions on Mathematical Software (TOMS)
Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book
Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book
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This paper evaluates the usability of the FEniCS Project for mantle convection simulations by numerical comparison to three established benchmarks. The benchmark problems all concern convection processes in an incompressible fluid induced by temperature or composition variations, and cover three cases: (i) steady-state convection with depth- and temperature-dependent viscosity, (ii) time-dependent convection with constant viscosity and internal heating, and (iii) a Rayleigh-Taylor instability. These problems are modeled by the Stokes equations for the fluid and advection-diffusion equations for the temperature and composition. The FEniCS Project provides a novel platform for the automated solution of differential equations by finite element methods. In particular, it offers a significant flexibility with regard to modeling and numerical discretization choices; we have here used a discontinuous Galerkin method for the numerical solution of the advection-diffusion equations. Our numerical results are in agreement with the benchmarks, and demonstrate the applicability of both the discontinuous Galerkin method and FEniCS for such applications.