Fictitious domain methods to solve convection-diffusion problems with general boundary conditions

International audienceSince a few years, fictitious domain methods have been arising for Computational Fluid Dynamics. The main idea of these methods consists in immersing the original physical domain in a geometrically bigger and simply-shaped other one called fictitious domain. As the spatial discretization is then performed in the fictitious domain, simple structured meshes can be used. The aim of this paper is to solve convection-diffusion problems with fictitious domain methods which can easily simulate free-boundary with possibly deformations of the boundary without increasing the computational cost. Two fictitious domain approaches performing either a spread interface or a thin interface are introduced. These two approaches require neither the modification of the numerical scheme near the immersed interface nor the use of Lagrange multipliers. Several ways to impose general embedded boundary conditions (Dirichlet, Robin or Neumann) are presented. The spread interface approach is computed using a finite element method as a finite volume method is used for the thin interface approach. The numerical schemes conserve the first- order accuracy with respect to the discretization step as observed in the numerical results reported here. The spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vicinity of the immersed boundary. The results obtained are full of promise, more especially as convection-diffusion equations are the core of the resolution of Navier-Stokes equations

Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods

We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement, and secondly to investigate the efficiency of the global error estimate

An improved method for solving quasilinear convection diffusion problems on a coarse mesh

A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The Newton-like iterations of the solver are based on the framework of regularized pseudo-transient continuation where the proposed time integrator is a variation on the Newmark strategy, designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite. An algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning. The method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization. Derivation and q-linear local convergence of the method is established, and numerical examples demonstrate the theory including the predicted rate of convergence of the iterations.Comment: 21 pages, 8 figures, 1 tabl

Implementation and application of adaptive mesh refinement for thermochemical mantle convection studies

Numerical modeling of mantle convection is challenging. Owing to the multiscale nature of mantle dynamics, high resolution is often required in localized regions, with coarser resolution being sufficient elsewhere. When investigating thermochemical mantle convection, high resolution is required to resolve sharp and often discontinuous boundaries between distinct chemical components. In this paper, we present a 2-D finite element code with adaptive mesh refinement techniques for simulating compressible thermochemical mantle convection. By comparing model predictions with a range of analytical and previously published benchmark solutions, we demonstrate the accuracy of our code. By refining and coarsening the mesh according to certain criteria and dynamically adjusting the number of particles in each element, our code can simulate such problems efficiently, dramatically reducing the computational requirements (in terms of memory and CPU time) when compared to a fixed, uniform mesh simulation. The resolving capabilities of the technique are further highlighted by examining plume‐induced entrainment in a thermochemical mantle convection simulation

Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp. 2228--2255] was restricted to symmetric problems, the present analysis also covers non-symmetric problems and hence the important case of present convection