In most of the method of fundamental solutions (MFS) approaches employed so far for the solution of inverse geometric problems, the MFS implementation typically leads to non-linear systems which were solved by standard nonlinear iterative least squares software. In the current approach, we apply a three-step non-iterative MFS technique for identifying a rigid inclusion from internal data measurements, which consists of: (i) a direct problem to calculate the solution at the set of measurement points, (ii) the solution of an ill-posed linear problem to determine the solution on a known virtual boundary and (iii) the solution of a direct problem in the virtual domain which leads to the identification of the unknown curve using the ${\rm MATLAB}^®$ functions contour in 2D and isosurface in 3D. The results of several numerical experiments for steady-state heat conduction and linear elasticity in two and three dimensions are presented and analyzed.

The dual consistency is an important issue in developing stable DWR error estimation towards the goal-oriented mesh adaptivity. In this paper, such an issue is studied in depth based on a Newton-GMG framework for the steady Euler equations. Theoretically, the numerical framework is redescribed using the Petrov-Galerkin scheme, based on which the dual consistency is depicted. It is found that for a problem with general configuration, a boundary modification technique is an effective approach to preserve the dual consistency in our numerical framework. Numerically, a geometrical multigrid is proposed for solving the dual problem, and a regularization term is designed to guarantee the convergence of the iteration. The following features of our method can be observed from numerical experiments, i). a stable numerical convergence of the quantity of interest can be obtained smoothly for problems with different configurations, and ii). towards accurate calculation of quantity of interest, mesh grids can be saved significantly using the proposed dual-consistent DWR method, compared with the dual-inconsistent one.

In this paper, a fifth-order hybrid multi-resolution weighted essentially non-oscillatory (WENO) scheme in the finite difference framework is proposed for solving one- and two-dimensional Hamilton-Jacobi equations. Firstly, a new discontinuity sensor is designed based on the extreme values of the highest degree polynomial in the multi-resolution WENO procedures. This hybrid strategy does not contain any human parameters related to specific problems and can identify the troubled grid points accurately and automatically. Secondly, a hybrid multi-resolution WENO scheme for Hamilton-Jacobi equations is developed based on the above discontinuity sensor and a simplified multi-resolution WENO scheme, which yields uniform high-order accuracy in smooth regions and sharply resolves discontinuities. Compared with the existing multi-resolution WENO scheme, the method developed in this paper can inherit its many advantages and is more efficient. Finally, some benchmark numerical experiments are performed to demonstrate the performance of the presented fifth-order hybrid multi-resolution WENO scheme for one- and two-dimensional Hamilton-Jacobi equations.

A second order accurate in time, finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the chemical potential, a modified Crank-Nicolson approximation is applied to the singular logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. Moreover, a nonlinear artificial regularization term is included in the chemical potential approximation, which ensures the positivity-preserving property for the logarithmic arguments, i.e., the numerical value of the phase variable is always between −1 and 1 at a point-wise level. Meanwhile, the convective term in the phase field evolutionary equation is updated in a semi-implicit way, with second order accurate temporal approximation. The fluid momentum equation is also computed by a semi-implicit algorithm. The unique solvability and the positivity-preserving property of the second order scheme is proved, accomplished by an iteration process. A modified total energy stability of the second order scheme is also derived. Some numerical results are presented to demonstrate the accuracy and the robust performance of the proposed second order scheme.

The high-order gas-kinetic scheme (HGKS) features good robustness, high efficiency and satisfactory accuracy,the performance of which can be further improved combined with WENO-AO (WENO with adaptive order) scheme for reconstruction. To reduce computational costs in the reconstruction procedure, this paper proposes to combine HGKS with a hybrid WENO-AO scheme. The hybrid WENO-AO scheme reconstructs target variables using upwind linear approximation directly if all extreme points of the reconstruction polynomials for these variables are outside the large stencil. Otherwise, the WENO-AO scheme is used. Unlike combining the hybrid WENO scheme with traditional Riemann solvers, the troubled cell indicator of the hybrid WENO-AO method is fully utilized in the spatial reconstruction process of HGKS. During normal and tangential reconstruction, the gas-kinetic scheme flux not only needs to reconstruct the conservative variables on the left and right interfaces but also to reconstruct the derivative terms of the conservative variables. By reducing the number of times that the WENO-AO scheme is used, the calculation cost is reduced. The high-order gas-kinetic scheme with the hybrid WENO-AO method retains original robustness and accuracy of the WENO5-AO GKS, while exhibits higher computational efficiency.

In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation in all ranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the small Knudsen number regime. The first APNN for linear transport equation is based on even-odd decomposition, which relaxes the stringent conservation prerequisites while concurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundary conditions is crucial for this network. For the Boltzmann-BGK equation, the APNN incorporates the conservation of mass, momentum, and total energy into the APNN framework as well as exact boundary conditions. A notable finding of this study is that approximating the zeroth, first, and second moments—which govern the conservation of density, momentum, and energy for the Boltzmann-BGK equation, is simpler than the distribution itself. Another interesting phenomenon observed in the training process is that the convergence of density is swifter than that of momentum and energy. Finally, we investigate several benchmark problems to demonstrate the efficacy of our proposed APNN methods.

An implicit, asymptotic-preserving and energy-charge-conserving (APECC) Particle-In-Cell (PIC) method is proposed to solve the Vlasov-Maxwell (VM) equations in the quasi-neutral regime. Charge conservation is enforced by particle orbital averaging and fixed sub-time steps. The truncation error depending on the number of sub-time steps is further analyzed. The temporal discretization is chosen by the Crank-Nicolson method to conserve the discrete energy exactly. The key step in the asymptotic-preserving iteration for the nonlinear system is based on a decomposition of the current density deduced from the Vlasov equation in the source of the Maxwell model. Moreover, we show that the convergence is independent of the quasi-neutral parameter. Extensive numerical experiments show that the proposed method can achieve asymptotic preservation and energy-charge conservation.

Deep Ritz method is a deep learning paradigm to solve partial differential equations. In this article we study the generalization error of the Deep Ritz method. We focus on the quintessential problem which is the Poisson’s equation. We show that generalization error of the Deep Ritz method converges to zero with rate $\frac{C}{\sqrt{n}},$ and we discuss about the constant $C.$ Results are obtained for shallow and residual neural networks with smooth activation functions.

The aim of this paper is to propose and analyze two postprocessing techniques for improving the accuracy of the $C^1$- and $C^0$-continuous Galerkin (CG) time stepping methods for nonlinear second-order initial value problems, respectively. We first derive several optimal a priori error estimates and nodal superconvergent estimates for the $C^1$- and $C^0$-$CG$ methods. Then we propose two simple but efficient local postprocessing techniques for the $C^1$- and $C^0$-$CG$ methods, respectively. The key idea of the postprocessing techniques is to add a certain higher order generalized Jacobi polynomial of degree $k+1$ to the $C^1$- or $C^0$-$CG$ approximation of degree $k$ on each local time step. We prove that, for problems with regular solutions, such postprocessing techniques improve the global convergence rates for the $L^2$-, $H^1$- and $L^∞$-error estimates of the $C^1$- and $C^0$-$CG$ methods with quasi-uniform meshes by one order. As applications, we apply the superconvergent postprocessing techniques to the $C^1$- and $C^0$-$CG$ time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.