Environmental DNA (eDNA) techniques are gaining attention as cost-effective, non-invasive strategies for acquiring information on fish and other aquatic organisms from water samples. Currently, eDNA approaches are used to detect specific fish species and determine fish community diversity. Various protocols used with eDNA methods for aquatic organism detection have been reported in different eDNA studies, but there are no general recommendations for fish detection. Herein, we reviewed 168 papers to supplement and highlight the key criteria for each step of eDNA technology in fish detection and provide general suggestions for eliminating detection errors. Although there is no unified recommendation for the application of diverse eDNA in detecting fish species, in most cases, 1 or 2 L surface water collection and eDNA capture on 0.7-μm glass fiber filters followed by extraction with a DNeasy Blood and Tissue Kit or PowerWater DNA Isolation Kit are useful for obtaining high-quality eDNA. Subsequently, species-specific quantitative polymerase chain reaction (qPCR) assays based on mitochondrial cytochrome b gene markers or eDNA metabarcoding based on both 12S and 16S rRNA markers via high-throughput sequencing can effectively detect target DNA or estimate species richness. Furthermore, detection errors can be minimized by mitigating contamination, negative control, PCR replication, and using multiple genetic markers. Our aim is to provide a useful strategy for fish eDNA technology that can be applied by researchers, advisors, and managers.

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In this paper, we introduce a general multilevel gradient Uzawa algorithm for symmetric saddle point systems. We compare its performance with the performance of the standard Uzawa multilevel algorithm. The main idea of the approach is to combine a double inexact Uzawa algorithm at the continuous level with a gradient type algorithm at the discrete level. The algorithm is based on the existence of a priori multilevel sequences of nested approximation pairs of spaces, but the family does not have to be stable. To ensure convergence, the process has to maintain an accurate representation of the residuals at each step of the inexact Uzawa algorithm at the continuous level. The residual representations at each step are approximated by projections or representation operators. Sufficient conditions for ending the iteration on a current pair of discrete spaces are determined by computing simple indicators that involve consecutive iterations. When compared with the standard Uzawa multilevel algorithm, our proposed algorithm has the advantages of automatically selecting the relaxation parameter, lowering the number of iterations on each level, and improving on running time. By carefully choosing the discrete spaces and the projection operators, the error for the second component of the solution can be significantly improved even when comparison is made with the discretization on standard families of stable pairs.