Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$ computational work, where $p$ is the degree of basis polynomials used, and $d$ is the spatial dimension. Our SVD-based tensor-product preconditioner requires $\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in $p$ per degree of freedom in 2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to $\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to $p=15$. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees $p$.Comment: 40 pages, 15 figure

Uniform subspace correction preconditioners for discontinuous Galerkin methods with hphp-refinement

In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with $hp$-refinement. These preconditioners are based on the decomposition of the DG finite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refined technique, which in this work we extend to the $hp$-refinement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh $h$, and the degree of polynomial approximation $p$. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refined meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).Comment: 24 pages, 9 figure

Evaluating Biophysical Attributes of Environmentally Degraded Landscapes in Northern Ethiopia using LANDSAT ETM data and GIS

Biophysical attributes of environmentally degraded landscapes in Adwa district, northern Ethiopia, were evaluated using LANDSAT ETM data and GIS. Satellite remote sensing (RS) has captured the spatial distribution and variability of Adwa land covers (75% classification accuracy, 73% Kappa statistic). GIS-based analysis of degraded land’s biophysical attributes has revealed associations between land-cover types, landform elements and major soils groups in the district. Agricultural farms were located closer to human settlements, while woodlands furthest away from settlements. Moreover, wooded croplands were found between arable and woodlands, indicating encroaching human activities through agricultural expansion. Forests and woodlands were dominant on high mountains, steep slopes and depressions, while degraded shrublands and scrublands were prominent on Leptosols and on dissected uplands and hills. On the other hand, agriculture was prominent on rolling hills and uplands, concave-shaped foot-slopes, and on the soils of the district characterized as Fluvisols and Vertisols. This study provides base-line information and add to land cover knowledge for this and similar regions. Additionally, it has identified associations among biophysical attributes in degraded Ethiopian highlands have important management implications for both under-developed and over-utilized areas.Keywords: Land cover, land degradation, soil erosion, land rehabilitation, deforestation, Ethiopian highland