查看更多>>摘要：This article presents a new primal-dual weak Galerkin (PDWG) finite element method for transport equations in non-divergence form. The PDWG method employs locally reconstructed differential operators and stabilizers in the weak Galerkin framework, and yields a symmetric discrete linear system involving the primal variable and the dual variable (known as the Lagrangian multiplier) for the adjoint equation. Optimal order error estimates are established in various discrete Sobolev norms for the corresponding numerical solutions. Numerical results are reported to illustrate the accuracy and efficiency of the new PDWG method. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：We construct a family of embedded pairs for optimal explicit strong stability preserving Runge-Kutta methods of order 2 <= p <= 4 to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective property, large stability region, and small error values as defined in Dekker and Verwer (1984) and Kennedy et al. (2000). The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size. Through several numerical experiments, we assess the overall effectiveness in terms of work versus precision while also taking into consideration accuracy and stability. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
查看更多>>摘要：Borrvall and Petersson (2003) developed the first model for the topology optimization of fluids in Stokes flow. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this work, we prove novel regularity results and extend their numerical analysis. In particular, given an isolated local minimizer to the infinite-dimensional problem, we show that there exists a sequence of finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to it. We also provide the first numerical investigation into convergence rates. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：In this paper, we propose, analyze, and numerically validate an adaptive finite element method for two-dimensional time-harmonic magnetic induction intensity equations as well as their Perfectly Matched-Layer (PML) equations. Based on Hodge decomposition, the equations are transformed into scalar elliptic boundary value problems and numerically solved by using the P-1 finite element method. Also, we can solve another basic quantity E concerned in physics. A posterior error indicator based on a superconvergent functional value recovery is considered for the two kinds of equations. Numerical experiments are presented to illustrate the effectiveness of the a posterior error indicator and the corresponding adaptive algorithm. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：We hybridized the rigorous coupled-wave approach (RCWA) with transformation optics to develop a hybrid coordinate-transform method for solving the time-harmonic Maxwell equations in a 2D domain containing a surface-relief grating. In order to prove that this method converges for the p-polarization state, we studied several different but related scattering problems. The imposition of generalized non-trapping conditions allowed us to prove a-priori estimates for these problems. To do this, we proved a Rellich identity and used density arguments to extend the estimates to more general problems. These a-priori estimates were then used to analyze the hybrid method. We obtained convergence rates with respect to two different parameters, the first being a slice thickness indicative of spatial discretization in the depth dimension, the second being the number of terms retained in the Rayleigh-Bloch expansions of the electric and magnetic field phasors with respect to the other dimension. Testing with a numerical example revealed faster convergence than our analysis predicted. The hybrid method does not suffer from the Gibbs phenomenon seen with the standard RCWA. (C) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper deals with numerical methods for solving systems of nonlinear parabolic problems. Block monotone iterative methods, based on the Jacobi and Gauss-Seidel methods, are in use for solving nonlinear difference schemes which approximate the systems of nonlinear parabolic problems. In the view of the method of upper and lower solutions, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence and uniqueness of solutions. Constructions of initial upper and lower solutions are discussed. Numerical experiments are presented.(C) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：Traditional rational motion design describes separately the translation of a reference point in a body and the rotation of the body about it. This means that there is dependence upon the choice of reference point. When considering the derivative of a motion, some approaches require the transform to be unitary. This paper resolves these issues by establishing means for constructing free-form motions from specified control poses using multiplicative and additive approaches. It also establishes the derivative of a motion in the more general non-unitary case. This leads to a characterization of the motion at the end of a motion segment in terms of the end pose and the linear and angular velocity and this, in turn, leads to the ability to join motion segments together with either C-1- or G(1)-continuity. (C) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually computationally "forbidden" combination of poorly conditioned equation system matrices due to challenging geometries, optimal control searches with iterative methods, slow convergence to system solutions on deterministic and non-deterministic level, and expensive remeshing due to geometrical changes. We overcome all these difficulties, utilizing the advantages of proper preconditioners adapted to unfitted mesh methods, improved types of Monte Carlo methods, and mainly employing the advantages of embedded FEMs, based on a fixed background mesh computed once even if geometrical changes are taking place. The sensitivity of the control problem is introduced in terms of random domains, employing a Quasi Monte Carlo method and emphasizing on a deterministic target state. The variational discretization concept is adopted, optimal error estimates for the state, adjoint state and control are derived that confirm the efficiency of the cut finite element method in challenging geometries. The performance of a multigrid scheme especially developed for unfitted finite element discretizations adapted to the optimal control problem is also tested. Some fundamental preconditioners are applied to the arising sparse linear systems coming from the discretization of the state and adjoint state variational forms in the spatial domain. The corresponding convergence rates along with the quality of the prescribed preconditioners are verified by numerical examples. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we construct two goal-oriented a posteriori error estimates with finite element approximations in the enriched space for the reaction-diffusion equation with the nonlinear reaction term. The first error estimator is proved to be the rigorous global lower and upper bounds on the error in the quantity of interest, and the second one is computed simply and efficiently. We propose the corresponding goal-oriented adaptive finite element algorithms, and show the effectiveness and the similar performance of those methods with a series of numerical experiments. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks for each sub-domain. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interfaces. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems. (c) 2022 Elsevier B.V. All rights reserved.
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