Multigrid waveform relaxation for the time-fractional heat equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(N M \log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented

On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method

Robust preconditioners for a new stabilized discretization of the poroelastic equations

In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [45]. The discretization is proved to be well-posed with respect to the physical and discretization parameters, and thus provides a framework to develop preconditioners that are robust with respect to such parameters as well. We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization that leads to a smaller overall problem after static condensation. Numerical tests for both two- and three-dimensional problems confirm the robustness of the block preconditioners with respect to the physical and discretization parameters

Do Organic Livestock Farms Differ from Low-Input Conventional Ones? Insights Based on Beef Cattle in Southern Europe

The objective of this study is to determine whether there are clear differences between conventional (but low-input) and organic beef cattle farms located in the Southwest of Spain. Thirty-three conventional and 30 organic farms were compared in terms of structure, technical management, and performance. The results showed that organic farms (‘All Organic’) mainly focus on the production of calves at weaning age, which are fattened in conventional holdings (‘Organic 1’; n = 22). The remaining organic farms (‘Organic 2’; n = 11) showed to participate in almost all stages of the agri-value chain. ‘Conventional’ farms were mainly dedicated to producing calves at weaning age (similarly to Organic 1). Organic 1 had the smallest herd size (80.18 livestock units (LU), p < 0.05). Organic 2 showed greater presence of indigenous breeds (62.08%, p < 0.05). Conventional farms proved to bear higher feed and veterinary costs per area (161.59 and 17.87 €/ha; p < 0.01 and p < 0.05, respectively), but Organic 2 had higher feed costs per LU. Therefore, Conventional and All Organic were quite similar, and differences depended mainly on farm structure. Hence, being either conventional or organic does not seem to be a valid criterion for drawing conclusions regarding the benefits or characteristics of each system