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The surface waves investigation and monitoring (SWIM) instrument onboard the China-France Oceanography Satellite (CFOSAT) can retrieve directional wave spectra with a wavelength range of 70-500 m. This study aims to validate the partitioned integrated wave parameters (PIWPs) from SWIM, including partitioned significant wave height (PSWH), partitioned peak wave period (PPWP), and partitioned peak wave direction (PPWD), against those from National Data Buoy Center (NDBC) buoys. With quasi-simultaneous spectra from two NDBC buoys 13 km away from each other near Hawaii, the methods of comparing PIWPs from two sets of spectra were discussed first. After cross-assigning partitions according to the spectral distance, it is found that wrong cross-assignments lead to many outliers strongly impacting the estimate of error metrics. Three methods, namely comparing only the best-matched partition, changing the threshold of spectral distance during cross-assignment, and maximum likelihood estimation of root-mean-square error (RMSE) of PIWPs, were used to reduce the impact of potential wrong cross-assignments. Using these methods, the SWIM PIWPs were validated against NDBC buoys. The results show that SWIM performs well at finding the spectral peaks of different partitions with the RMSE of PPWPs and PPWDs of 0.9 s and 20°, respectively, which can be a useful complement for other wave observations. However, the accuracy of PSWH from SWIM is not that good at this stage, probably because the high noise level in the spectra impacts the result of the partitioning algorithm. Further improvement is needed to obtain better PSWH information.

In this paper, we propose a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence on polytopal mesh in the primary velocity-pres...

In this paper, we develop a new WENO weak Galerkin finite element scheme for solving the time dependent hyperbolic equations. The upwind-type stabilizer is imposed to enforce the flux direction in the scheme. For the linear convection equations, we analyze the L 2 -stability and error estimate for L 2 -norm. We also investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for obtaining a robust procedure to achieve high order accuracy and capture the sharp, non-oscillatory shock transitions. The approach applies for linear convection equations and Burgers equations. Finally, numerical examples are presented for validating the theoretical conclusions.

Pressure-robustness is an essential demand for the incompressible fluid simulation. In this paper, we develop the enhanced discontinuous Galerkin (DG) finite element methods for solving Stokes equations in the primary velocity-pressure formulation to achieve pressure-robustness. The velocity reconstruction operator has been designed for discontinuous functions and utilized to modify the external source assembling. The new schemes stay almost the same as that in the existing DG schemes but only differ for the source terms. The conforming discontinuous Galerkin and symmetric interior penalty DG have been employed to demonstrate the enhancement. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Several numerical experiments are performed to validate our theoretical conclusions.

The 2019 novel coronavirus (2019-nCoV) outbreak has been treated as a Public Health Emergency of International Concern by the World Health Organization. This work made an early prediction of the 2019-nCoV outbreak in China based on a simple mathematical model and limited epidemiological data. Combing characteristics of the historical epidemic, we found part of the released data is unreasonable. Through ruling out the unreasonable data, the model predictions exhibit that the number of the cumulative 2019-nCoV cases may reach 76,000 to 230,000, with a peak of the unrecovered infectives (22,000-74,000) occurring in late February to early March. After that, the infected cases will rapidly monotonically decrease until early May to late June, when the 2019-nCoV outbreak will fade out. Strong anti-epidemic measures may reduce the cumulative infected cases by 40%-49%. The improvement of medical care can also lead to about one-half transmission decrease and effectively shorten the duration of the 2019-nCoV., The flow diagram of early prediction of the 2019-nCoV. The early-stage epidemiological data is firstly analyzed to rule out unreasonable part with considering historical epidemic. The SIR model is then simplified to reduce the number of parameters and thus data requirement. The early prediction is finally obtained via integrating the simplified model with parameter estimations based on objective analysis.

In this paper, we develop a pressure robust weak Galerkin finite element scheme for Stokes equations on polygonal mesh. The major idea for achieving a pressure-independent energy-error estimate is ...

In this paper, we develop a uniform robust weak Galerkin finite element scheme for Brinkman equations. The major idea for achieving uniform energy-error estimate is to use a divergence preserving v...

In this paper, we derive an a posteriori error estimator for the weak Galerkin least-squares (WG-LS) method applied to the reaction–diffusion equation. We show that this estimator is both reliable and efficient, allowing it to be used for adaptive refinement. Due to the flexibility of the WG-LS discretization, we are able to design a simple and straightforward refinement scheme that is applicable to any shape regular polygonal mesh. Finally, we present numerical experiments that confirm the effectiveness of the estimator, and demonstrate the robustness and efficiency of the proposed adaptive WG-LS approach.

In this paper, we present a posteriori error estimate of weak Galerkin (WG) finite element methods based on the second order elliptic interface problems. This estimate can be applied to polygonal meshes or meshes with hanging nodes. The reliability and efficiency of the designed error estimator has been proved in this work. Extensive numerical tests are performed to validate our algorithm. These results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator.

This paper introduces a weak Galerkin (WG) finite element method for the Navier–Stokes equations in the primal velocity–pressure formulation. Optimal-order error estimates are established for the corresponding numerical approximations. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular. Numerical experiments are presented to support the theoretical results.

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion-convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design., 24 pages, 17 figures

El Niño-Southern Oscillation (ENSO), which is one of the main drivers of Earth’s inter-annual climate variability, often causes a wide range of climate anomalies, and the advance prediction of ENSO is always an important and challenging scientific issue. Since a unified and complete ENSO theory has yet to be established, people often use related indicators, such as the Niño 3.4 index and southern oscillation index (SOI), to predict the development trends of ENSO through appropriate numerical simulation models. However, because the ENSO phenomenon is a highly complex and dynamic model and the Niño 3.4 index and SOI mix many low- and high-frequency components, the prediction accuracy of current popular numerical prediction methods is not high. Therefore, this paper proposed the ensemble empirical mode decomposition-temporal convolutional network (EEMD-TCN) hybrid approach, which decomposes the highly variable Niño 3.4 index and SOI into relatively flat subcomponents and then uses the TCN model to predict each subcomponent in advance, finally combining the sub-prediction results to obtain the final ENSO prediction results. Niño 3.4 index and SOI reanalysis data from 1871 to 1973 were used for model training, and the data for 1984–2019 were predicted 1 month, 3 months, 6 months, and 12 months in advance. The results show that the accuracy of the 1-month-lead Niño 3.4 index prediction was the highest, the 12-month-lead SOI prediction was the slowest, and the correlation coefficient between the worst SOI prediction result and the actual value reached 0.6406. Furthermore, the overall prediction accuracy on the Niño 3.4 index was better than that on the SOI, which may have occurred because the SOI contains too many high-frequency components, making prediction difficult. The results of comparative experiments with the TCN, LSTM, and EEMD-LSTM methods showed that the EEMD-TCN provides the best overall prediction of both the Niño 3.4 index and SOI in the 1-, 3-, 6-, and 12-month-lead predictions among all the methods considered. This result means that the TCN approach performs well in the advance prediction of ENSO and will be of great guiding significance in studying it.

We developed a domain decomposition model reduction method for linear steady- state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion ...

In this paper, we present a simple a posteriori error estimate for the weak Galerkin finite element method for the Stokes equation. This residual type estimator can be applied to general meshes suc...

A discrete divergence free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in [17] . Discrete divergence free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence free basis derived, the discrete divergence free WG scheme can eliminate pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence free WG method.

In this paper, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. This new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

We develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.

In this paper, we introduce a pressure-robust virtual element method for the Stokes problem on convex polygonal meshes. The method is based on the lowest order virtual element, in which the degrees of freedom for the velocity are simply given by the evaluations of velocity at the mesh vertices and the average values of normal velocity across the mesh edges, and the pressure is approximated by piecewise constants. In the standard virtual element scheme, an average of nodal values of test function is used in the approximation of right hand side to obtain the computability and achieve optimal approximation. However, such standard scheme involves a pressure contribution in the velocity error. To achieve the pressure-independent velocity approximation, we define an H ( div ) -conforming velocity reconstruction operator for the velocity test function and propose the modified scheme by employing it in the approximation of right-hand-side source term assembling. Compared with the standard scheme, the stiffness matrix keeps unchanged and only the approximation of the right hand side changes. The error estimates for the velocity and pressure have been proved, which imply that the velocity error is independent of pressure. Numerical experiments are shown to validate the theoretical conclusions.

Consider the Poisson equation in a polytopal domain $$\Omega \subset {\mathbb {R}}^d$$ ( $$d=2, 3$$ ) as the model problem. We study interior energy error estimates for the weak Galerkin finite element approximation to elliptic boundary value problems. In particular, we show that the interior error in the energy norm is bounded by three components: the best local approximation error, the error in negative norms, and the trace error on the element boundaries. This implies that the interior convergence rate can be polluted by solution singularities on the domain boundary, even when the solution is smooth in the interior region. Numerical results are reported to support the theoretical findings. To the best of our knowledge, this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes.

A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.

In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. A uniform numerical stability with respect to the Knudsen number $$\epsilon $$ , and a uniform in $$\epsilon $$ error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for deterministic problem. We provide a rigorous proof of the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.

Geometric and electrostatic modeling is an essential component in computational biophysics and molecular biology. Commonly used geometric representations admit geometric singularities such as cusps, tips and self-intersecting facets that lead to computational instabilities in the molecular modeling. The present work explores the use of flexibility and rigidity index (FRI), which has a proved superiority in protein B-factor prediction, for biomolecular geometric representation and associated electrostatic analysis. FRI rigidity surfaces are free of geometric singularities. We proposed a rigidity based Poisson–Boltzmann equation for biomolecular electrostatic analysis. Our approaches to surface and electrostatic modeling are validated by a set of 21 proteins. Our results are compared with those of established methods. Finally, being smooth and analytically differentiable, FRI rigidity functions offer excellent curvature analysis, which characterizes concave and convex regions on protein surfaces. Polarized curvatures constructed by using the product of minimum curvature and electrostatic potential is shown to predict potential protein–ligand binding sites.

In this paper, we develop a new upwind weak Galerkin finite element scheme for linear hyperbolic equations. The upwind-type stabilizer is imposed in the scheme. An error estimate is investigated for a suitable norm. Finally, numerical examples are presented for validating the theoretical conclusions.

In this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for both the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. The numerical examples cover a wide range of applied problems, including singularly...

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H 1 and L 2 norms are established for the present WG finite element solutions. Extensive numerical experiments have been conducted to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L ∞ norm for both C 1 and H 2 continuous solutions.

The Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

This paper studies a weak Galerkin finite element method on a domain with curved edges. We consider the two dimensional Poisson’s equation of a fixed curved boundary. Usually, the straight polygon meshes deliver a sub-optimal rate of convergence for the curved domain as approximation order bounded by O ( h 2 ) even for high order numerical schemes. In this paper, we shall develop a high order (optimal rate of convergence) WGFEM based on the curved mesh. The implementation details are studied for efficient numerical integration tools. Numerical experiments are provided to show the proposed curved WGFEM produces an optimal rate for errors measured in L 2 -norm and energy-norm of convergence as O ( h k + 1 ) and O ( h k ) for curved two-dimensional domain.

In this paper, we present a time-explicit sparse grid discontinuous Galerkin method for solving the three-dimensional time-domain Maxwell equations. The conservation properties and convergence rates are established for different choices of numerical fluxes. The convergence rates are proved theoretically and then verified by several numerical examples. Even though our scheme does not preserve the divergence, but by implying the higher order polynomial, one can observe the same convergence rate as numerical solution. Several numerical tests are presented to validate these conclusions.

In this paper, a general superconvergence result of H ( div ) finite element approximations for the Stokes equations is established by using the local L 2 -projection method. Local regularity assumptions for the Stokes problem on regular partitions are required in order to derive superconvergence conclusions. Several numerical experiments are given to verify the theoretical results.

In this work, the residual-type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in H1 norm and pressure in L2 norm are derived. Furthermore, a global upper bound of u − uh in L2-norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.

A modified weak Galerkin (MWG) finite element method is introduced for the Stokes equations in the primary velocity-pressure formulation. This method is based on the weak Galerkin (WG) finite element method developed recently in Wang and Ye (0000). The unknowns associated with element boundaries in the WG method have been eliminated in the MWG method. As the result, the MWG method has fewer degrees of freedom. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Numerical results are presented to confirm the theory.

In this paper, we present an immersed weak Galerkin method for solving second-order elliptic interface problems. The proposed method does not require solution meshes to be aligned with the interface. Consequently, uniform Cartesian meshes can be used for nontrivial interfacial geometry. We show the existence and uniqueness of the numerical algorithm, and provide the error analysis in the energy norm. Numerical results are reported to demonstrate the performance of the method.

In this paper, we develop a modified weak Galerkin finite element method on arbitrary grids for convection–diffusion problems in two dimensions based on our previous work (Wang et al., J Comput Appl Math 271, 319–327, 2014), in which we only considered second order Poisson equations and thus only introduced a modified weak gradient operator. This method, called MWG-FEM, is based on a modified weak gradient operator and weak divergence operator which is put forward in this paper. Optimal order error estimates are established for the corresponding MWG-FEM approximations in both a discrete $$H^1$$ norm and the standard $$L^2$$ norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the MWG-FEM.

This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors. We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow.

with homogenous Dirichlet boundary condition, where ut is the time partial derivative of u(x, t); a(x) is a uniformly positive on Ω and a(x), f(x, t) and u(x) are assumed to be sufficiently smooth. Since the 1950s, scientists have formulated time-stepping procedures to numerically approximate the solutions of such problems. Numerical methods for such parabolic problems can be classified as two categories. The first category consists of finite difference methods that use difference quotient to replace differential quotient and the other refers to as finite element methods, see, e.g., [3, 6, 7, 12–14, 18] and references in. The WG-FEMs, which was first introduced by Wang and Ye [16] for solving the second order elliptic problems, are newly developed FEMs. The novel idea of WG-FEMs is to introduce weak functions and weak derivatives, and allows the use of totally discontinuous piecewise polynomials in the finite element procedure. Later, The WG -FEMs were studied from implementation point of view in [8] and applied to solve the Helmholtz problem with high wave numbers in [10]. A WG-FEM was introduced and analyzed for parabolic equations based on a discrete weak gradient arising from local RT [11]. Due to the use of RT elements, the WG finite element

A weak Galerkin (WG) method is introduced and numerically tested for the Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers., Comment: 18 pages, 11 figures, 6 tables. arXiv admin note: substantial text overlap with arXiv:1111.0671

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding WG finite element solutions. Error estimates in the usual $L^2$ norm are also derived, yielding a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions., 23 pages, 1 figure, 2 tables. arXiv admin note: text overlap with arXiv:1202.3655, arXiv:1204.3655

This paper presents two types of a posterior error estimations for discontinuous finite volume methods to elliptic interface problems. One is the residual-based estimator and other one is the recovery-based estimator. For both a posterior error estimators, we establish reliability and efficiency bounds. Numerical experiments are conducted to validate our theoretical conclusions.

Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic partial differential equations (PDEs) with discontinuous coefficients and interfaces. The paper also presents many numerical tests for validating the WG-FEM for solving second order elliptic interface problems. For such interface problems, the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design high order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order one for the solution itself in $L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable of delivering numerical approximations that are of order 1.75 in the usual $L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high order of numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element., 24 pages, many figures from computation

In Cai and Zhang (2009, 2010) [12,13], they introduced the recovery-based a posteriori error estimator for conforming, mixed, and nonconforming finite element methods of interface problems. In this paper, we extend the idea to present a recovery-based a posterior error estimator for finite volume methods which employ the nonconforming linear trial functions to approximate elliptic interface problems. The method recovers the flux and gradient in H(div) and H(curl) conforming finite element spaces with a weighted L^2 projection, respectively. The reliability and efficiency bounds are established. Numerical experiments are given to support the conclusions.

A general superconvergence of discontinuous Galerkin (DG) finite element method for the elliptic problem is established by using L^2-projection method. Regularity assumptions for the elliptic problem with regular partitions are required. Numerical experiments are given to verify the theoretical results.

This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multi-component persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for protein-ligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 protein-ligand complexes from the PDBBind database and 128,374 ligand-target and decoy-target pairs in the DUD database are performed to test respectively the scoring power and the discriminatory power of the proposed topological learning strategies. It is demonstrated that the present topological learning outperforms other existing methods in protein-ligand binding affinity prediction and ligand-decoy discrimination., Author summary Conventional persistent homology neglects chemical and biological information during the topological abstraction and thus has limited representational power for complex chemical and biological systems. In terms of methodological development, we introduce advanced persistent homology approaches for the characterization of small molecular structures which can capture subtle structural difference. We also introduce electrostatic persistent homology to embed physics in topological invariants. These approaches encipher physics, chemistry and biology, such as hydrogen bonds, electrostatics, van der Waals interactions, hydrophobicity and hydrophilicity, into topological fingerprints which, although cannot literally recast into physical interpretations, are ideally suitable for machine learning, particularly deep learning, rendering topological learning algorithms. In terms of applications, we construct a structure-based virtual screening model which outperforms other existing methods. This competitive model on the DUD database is derived by assessing the performance of a comprehensive collection of topological approaches proposed in this work and introduced in our earlier work, on the PDBBind database. The topological features constructed in this work can readily be applied to other biomolecular problems where the characterization of proteins or small molecules is needed.

A general superconvergence result of H(div) finite element approximations for the Stokes equations is established by using L^2-projection method. Regularity assumptions for the Stokes problem with regular partitions is required. Numerical experiments are given to verify the theoretical results.

In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy model is discussed in great detail. Mathematically, unlike the previous persistent entropy or topological entropy, a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our multiscale persistent entropy can be used in conformation entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of a multiscale rigidity function built from bond and/or dihedral angle distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the mediate range. Moreover, our method can be successfully used in protein classification. For a test database with around nine hundred proteins, a clear separation between all-alpha and all-beta proteins can be achieved, using only the dihedral and pseudo-bond angle information. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Essentially, PSI can be used to describe the "regularity" information in any systems., 10 figures and 1 table

The weak Galerkin (WG) methods have been introduced in the references [11, 16] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique., 10 pages

This paper presents an error estimate in the L 2 -norm for the discontinuous Galerkin finite element methods (DGFEM) for elliptic problems with low regularity solutions. The Raviart–Thomas interpolation operator is employed to derive the new result, which complements the mesh-dependent energy norm error estimates in Gudi (2010) [2] . Numerical results corroborate the theoretical analysis.

This paper investigates convergence of the discontinuous finite volume method (DFVM) under minimal regularity assumptions on solutions of second order elliptic boundary value problems. Conventional analysis requires the solutions to be in Sobolev spaces H1+s,s>12. Here we assume the solutions are in H1+s,s>0 and employ the techniques developed in Gudi (2010) [18,20] to derive error estimates in a mesh-dependent energy norm and the L2-norm for DFVM. The theoretical estimates are illustrated by numerical results, which include problems with corner singularity and intersecting interfaces.