On locking and robustness in the finite element method
SIAM Journal on Numerical Analysis
Solving diffusion equations with rough coefficients in rough grids
Journal of Computational Physics
The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials
Journal of Computational Physics
Journal of Computational Physics
Finite Element Approximation of the Diffusion Operator on Tetrahedra
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A finite volume method for the approximation of diffusion operators on distorted meshes
Journal of Computational Physics
ACM Transactions on Mathematical Software (TOMS)
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
An Unsymmetrized Multifrontal LU Factorization
SIAM Journal on Matrix Analysis and Applications
A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation
Journal of Computational Physics
Journal of Computational and Applied Mathematics - Proceedings of the international conference on recent advances in computational mathematics
SIAM Journal on Numerical Analysis
Numerical Approximation of Partial Differential Equations
Numerical Approximation of Partial Differential Equations
A unified treatment of boundary conditions in least-square based finite-volume methods
Computers & Mathematics with Applications
Mathematics and Computers in Simulation
The Discrete Duality Finite Volume Method for Convection-diffusion Problems
SIAM Journal on Numerical Analysis
A nine-point scheme with explicit weights for diffusion equations on distorted meshes
Applied Numerical Mathematics
Bad behavior of Godunov mixed methods for strongly anisotropic advection-dispersion equations
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Monotonic solution of heterogeneous anisotropic diffusion problems
Journal of Computational Physics
Journal of Computational Physics
Mimetic finite difference method
Journal of Computational Physics
Hi-index | 31.47 |
Strongly anisotropic diffusion equations require special techniques to overcome or reduce the mesh locking phenomenon. We present a finite volume scheme that tries to approximate with the best possible accuracy the quantities that are of importance in discretizing anisotropic fluxes. In particular, we discuss the crucial role of accurate evaluations of the tangential components of the gradient acting tangentially to the control volume boundaries, that are called into play by anisotropic diffusion tensors. To obtain the sought characteristics from the proposed finite volume method, we employ a second-order accurate reconstruction scheme which is used to evaluate both normal and tangential cell-interface gradients. The experimental results on a number of different meshes show that the scheme maintains optimal convergence rates in both L^2 and H^1 norms except for the benchmark test considering full Neumann boundary conditions on non-uniform grids. In such a case, a severe locking effect is experienced and documented. However, within the range of practical values of the anisotropy ratio, the scheme is robust and efficient. We postulate and verify experimentally the existence of a quadratic relationship between the anisotropy ratio and the mesh size parameter that guarantees optimal and sub-optimal convergence rates.