The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain, here constructed from a level-set (signed distance) function, and source terms are introduced to approximate the boundary conditions. The reformulated equation can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in $\epsilon$, the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, $\epsilon\propto h$, the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in $\epsilon$, e.g., $O(\epsilon^2)$ in the \(L^2\) norm and $O(\epsilon^p)$ with $1.5\le p\le 2$ in the \(L^{\infty}\) norm. Our analytic results are confirmed numerically in stationary and moving domains. Finally, we extend our results to two-phase fluid problems and propose a diffuse domain approach to simulate two-phase flows in complex geometries.

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Rbfox1 is a splicing regulator that has been associated with various neurological conditions such as autism spectrum disorder, mental retardation, epilepsy, attention-deficit/hyperactivity disorder and schizophrenia. We show that in adult rodent retinas, Rbfox1 is expressed in all types of retinal ganglion cells (RGCs) and in certain subsets of amacrine cells (ACs), within the inner nuclear (INL) and ganglion cell (GCL) layers. In the INL, all Rbfox1-positive cells were colocalized with GABAergic ACs, however not all GABAergic ACs were immunostained for Rbfox1. In the GCL, a vast majority of GABAergic dACs were Rbfox1-immunopositive. Furthermore, all cholinergic starburst ACs (SACs) in the INL (type a) and in the GCL (type b) were Rbfox1 positive. The expression of Rbfox1 in the retina significantly overlapped with expression of Rbfox2, another member of Rbfox family of proteins. Rbfox2, in addition to RGCs and ACs, was also expressed in horizontal cells. In developing retinas at E12 and E15, Rbfox1 is localized to the cytoplasm of differentiating RGCs and ACs. Between P0 and P5, Rbfox1 subcellular localization switched from cytoplasmic to predominantly nuclear. Downregulation of Rbfox1 in adult Rbfox1loxP/loxP mice had no detectable effect on retinal gross morphology. However, the visual cliff test revealed marked abnormalities of depth perception of these animals. RNA sequencing of retinal transcriptomes of control and Rbfox1 knockout animals identified a number of Rbfox1-regulated genes that are involved in establishing neuronal circuits and synaptic transmission, including Vamp1, Vamp2, Snap25, Trak2, and Slc1A7, suggesting the role of Rbfox1 in facilitating synaptic communications between ACs and RGCs.

Rbfox1 is a splicing regulator that has been associated with various neurological conditions such as autism spectrum disorder, mental retardation, epilepsy, attention-deficit/hyperactivity disorder and schizophrenia. We show that in adult rodent retinas, Rbfox1 is expressed in all types of retinal ganglion cells (RGCs) and in certain subsets of amacrine cells (ACs), within the inner nuclear (INL) and ganglion cell (GCL) layers. In the INL, all Rbfox1-positive cells were colocalized with GABAergic ACs, however not all GABAergic ACs were immunostained for Rbfox1. In the GCL, a vast majority of GABAergic dACs were Rbfox1-immunopositive. Furthermore, all cholinergic starburst ACs (SACs) in the INL (type a) and in the GCL (type b) were Rbfox1 positive. The expression of Rbfox1 in the retina significantly overlapped with expression of Rbfox2, another member of Rbfox family of proteins. Rbfox2, in addition to RGCs and ACs, was also expressed in horizontal cells. In developing retinas at E12 and E15, Rbfox1 is localized to the cytoplasm of differentiating RGCs and ACs. Between P0 and P5, Rbfox1 subcellular localization switched from cytoplasmic to predominantly nuclear. Downregulation of Rbfox1 in adult Rbfox1loxP/loxP mice had no detectable effect on retinal gross morphology. However, the visual cliff test revealed marked abnormalities of depth perception of these animals. RNA sequencing of retinal transcriptomes of control and Rbfox1 knockout animals identified a number of Rbfox1-regulated genes that are involved in establishing neuronal circuits and synaptic transmission, including Vamp1, Vamp2, Snap25, Trak2, and Slc1A7, suggesting the role of Rbfox1 in facilitating synaptic communications between ACs and RGCs.

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