scholarly journals TWO-GRID WEAK GALERKIN METHOD FOR SEMILINEAR ELLIPTIC DIFFERENTIAL EQUATIONS

2022 ◽  
Author(s):  
luoping chen ◽  
fanyun wu ◽  
guoyan zeng
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Mesh Size ◽  
Initial Guess ◽  
Coarse Mesh ◽  
Weak Galerkin ◽  
Fine Mesh ◽  
Semilinear Elliptic ◽  
Elliptic Differential Equations ◽  
Linearized System

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

scholarly journals An Adaptive Approach to Solutions of Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Nebiye Korkmaz ◽  
Zekeriya Güney
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Iteration Method ◽  
Legendre Polynomials ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Coarse Mesh ◽  
Fine Mesh

As an approach to approximate solutions of Fredholm integral equations of the second kind, adaptive hp-refinement is used firstly together with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution. The linear hat functions and modified integrated Legendre polynomials are used as basis functions for the approximations. The most appropriate refinement is determined by an optimization problem given by Demkowicz, 2007. During the calculationsL2-projections of approximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated. Depending on the error values, these procedures could be repeated consecutively or different meshes could be used in order to decrease the error values.

EFFECTIVENESS OF MALAISE TRAPS IN COLLECTING HYMENOPTERA: THE INFLUENCE OF TRAP DESIGN, MESH SIZE, AND LOCATION

1988 ◽  
Vol 120 (8-9) ◽  
pp. 787-796 ◽  
Author(s):  
D. Christopher Darling ◽  
Laurence Packer
Keyword(s):  
Paired Comparison ◽  
Mesh Size ◽  
Graphical Analysis ◽  
Malaise Trap ◽  
Coarse Mesh ◽  
Trap Design ◽  
Fine Mesh ◽  
Provincial Park ◽  
Pan Traps ◽  
Malaise Traps

AbstractThe influence of various aspects of Malaise trap design on efficiency in collecting various groups of Hymenoptera was compared using commercially available traps. The influence of fine and coarse mesh sizes and the importance of the incorporation of pan traps into the design were evaluated in two sampling periods in an old dune community at Pinery Provincial Park in southern Ontario. Numbers of individuals collected in Malaise trap heads and pan traps were presented for each paired comparison of mesh size (fine and coarse) and location (top and bottom of a hill) for various families of Hymenoptera. A graphical analysis of the normalized catch data for more inclusive groupings, viz. Symphyta, Aculeata, Ichneumonoidea, and microhymenoptera, was presented. The major results of these comparisons were as follows: (1) pan traps were an important component of efficient Malaise traps, especially for Aculeata and microhymenoptera; (2) coarse mesh was more effective in collecting Aculeata; (3) coarse and fine mesh were both effective in collecting Ichneumonoidea; and (4) fine mesh was more effective in collecting microhymenoptera. There was an interaction effect between the type of trap used and groups collected and it was not possible to maximize simultaneously the collection of all groups of Hymenoptera. The use of various mesh types and a trap design that incorporates pan traps was recommended.

scholarly journals A Two-Level Additive Schwarz Preconditioning Algorithm for the Weak Galerkin Method for the Second-Order Elliptic Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Fangfang Qin ◽  
Min Zha ◽  
Feng Wang
Keyword(s):  
Elliptic Equation ◽  
Transfer Operator ◽  
Galerkin Approximation ◽  
Second Order ◽  
Coarse Mesh ◽  
Order Elliptic Equation ◽  
Weak Galerkin ◽  
Additive Schwarz ◽  
Fine Mesh ◽  
Second Order Elliptic Equation

This paper proposes a two-level additive Schwarz preconditioning algorithm for the weak Galerkin approximation of the second-order elliptic equation. In the algorithm, aP1conforming finite element space is defined on the coarse mesh, and a stable intergrid transfer operator is proposed to exchange the information between the spaces on the coarse mesh and the fine mesh. With the framework of the Schwarz method, it is proved that the condition number of the preconditioned system only depends on the rate of the coarse mesh size and the overlapping size. Some numerical experiments are carried out to verify the theoretical results.

scholarly journals Post-fire consequences for leaf breakdown in a tropical stream

2019 ◽  
Vol 31 ◽  
Author(s):  
Renan de Souza Rezende ◽  
Cristiano Queiroz de Albuquerque ◽  
Andrezza Sayuri Victoriano Hirota ◽  
Paulo Fernandes Roges Souza Silva ◽  
Ricardo Keichi Umetsu ◽  
...  
Keyword(s):  
Leaf Litter ◽  
Fungal Biomass ◽  
Mesh Size ◽  
Fire Effects ◽  
Dry Mass ◽  
Tropical Stream ◽  
Fire Event ◽  
Litter Bags ◽  
Fine Mesh ◽  
Post Fire Effects

Abstract Aim Wildfire is a natural pulsed disturbance in landscapes of the Savannah Biome. This study evaluates short-term post-fire effects on leaf litter breakdown, the invertebrate community and fungal biomass of litter from three different vegetal species in a tropical stream. Methods Senescent leaves of Inga laurina, Protium spruceanum and Rircheria grandis (2 ± 0.1 g dry mass) were individually placed in litter bags (30 × 30 cm: 10 mm coarse mesh and 0.5 mm fine mesh) and submerged in the study stream before and after fire. Replicate bags (n = 4; individually for each species, sampling time, fire event and mesh size) were then retrieved after 20 and 40 days and washed to separate the invertebrates before fire event and again immediately after fire. Disks were cut from leaves to determine ash-free dry mass, while the remaining material was oven-dried to determine dry mass. Results The pre-fire mean decomposition coefficient (k = -0.012 day-1) was intermediate compared to that reported for other savannah streams, but post-fire it was lower (k = -0.007 day-1), due to decreased allochthonous litter input and increased autochthones production. Intermediate k values for all qualities of litter post-fire may indicate that fire is equalizing litter quality in the stream ecosystem. The abundance of scrapers was found to be more important than fungal biomass or shredder abundance, probably due to their functioning in leaf fragmentation while consuming periphyton growing on leaf litter. Conclusions Theses results indicate that fire can modify the relationships within decomposer communities in tropical stream ecosystems.

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