Journal of Scientific Computing最新文献

A new discontinuous Galerkin (DG) method is proposed and analyzed for general second-order elliptic problems. It features that local (L^2) projections are used to reconstruct the diffusion term and the convection term and that it does not need any penalty and even does not need any stabilization in the formulation. The Babus̆ka inf-sup stability is proven. The error estimates are established. More importantly, the new DG method can hold the SUPG-type stability for the convection; the SUPG-type optimal error estimates ({{mathcal {O}}}(h^{ell +1/2})) is obtained for the problem with a dominating convection for the (ell )-th order ((ell ge 0)) discontinuous element. Numerical results are provided.

A combined hybrid mixed and hybridizable discontinuous Galerkin method is formulated for the flow and transport equations. Convergence of the method is obtained by deriving optimal a priori error bounds in the L(^2) norm in space. Since the velocity in the transport equation depends on the flow problem, the stabilization parameter in the HDG method is a function of the discrete velocity. In addition, a key ingredient in the convergence proof is the construction of a projection that is shown to satisfy optimal approximation bounds. Numerical examples confirm the theoretical convergence rates and show the efficiency of high order discontinuous elements.

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the (ell _0) norm. Specifically, the (ell _0) model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the (ell _0) norm regularization term. Such an (ell _0) model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the (ell _0) models found by the proposed inexact algorithm outperform global minimizers of the corresponding (ell _1) models, in terms of approximation accuracy and sparsity of the solutions.

This article presents a parallel finite element discretization scheme for solving numerically the steady natural convection equations, where a fully overlapping domain decomposition technique is used for parallelization. In this scheme, each processor computes independently a local solution in its subdomain using a mesh that covers the entire domain. It has a small mesh size h around the subdomain and a large mesh size H away from the subdomain. The discretization scheme is easy to implement based on existing serial software. It can yield an optimal convergence rate for the approximate solutions with suitable algorithmic parameters. Compared with the standard finite element method, the scheme is able to obtain an approximate solution of comparable accuracy with considerable reduction in computational time. Theoretical and numerical results show the promise of the scheme, where numerical simulation results for some benchmark problems such as the buoyancy-driven square cavity flow, right-angled triangular cavity flow and sinusoidal hot cylinder flow are provided.

This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an (H(textrm{div}))-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier–Stokes equations.

The main objective of this study is to evaluate the performance of serendipity virtual element methods in solving semilinear parabolic integro-differential equations with variable coefficients. The primary advantage of this method, in comparison to the standard (enhanced) virtual element methods, lies in the reduction of internal-moment degrees of freedom, which can speed up the iterative algorithms when using the quasi-interpolation operators to approximate nonlinear terms. The temporal discretization is obtained with the backward-Euler scheme. To maintain consistency with the accuracy order of the backward-Euler scheme, the integral term is approximated using the left rectangular quadrature rule. Within the serendipity virtual element framework, we introduced a Ritz–Volterra projection and conducted a comprehensive analysis of its approximation properties. Building upon this analysis, we ultimately provided optimal (H^1)-seminorm and (L^2)-norm error estimates for both the semi-discrete and fully discrete schemes. Finally, two numerical examples that serve to illustrate and validate the theoretical findings are presented.

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation on convex domains. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix–Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart–Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.

A singularly perturbed reaction–diffusion problem in 1D is solved numerically by a local discontinuous Galerkin (LDG) finite element method. For this type of problem the standard energy norm is too weak to capture the contribution of the boundary layer component of the true solution, so balanced norms have been used by many authors to give more satisfactory error bounds for solutions computed using various types of finite element method. But for the LDG method, up to now no optimal-order balanced-norm error estimate has been derived. In this paper, we consider an LDG method with central numerical flux on a Shishkin mesh. Using the superconvergence property of the local (L^2) projector and some local coupled projections around the two transition points of the mesh, we prove an optimal-order balanced-norm error estimate for the computed solution; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, in the balanced norm we establish (O((N^{-1}ln N)^{k+1})) convergence when k is even and (O((N^{-1}ln N)^{k})) when k is odd. Numerical experiments confirm the sharpness of these error bounds.

We present an algorithm for fast generation of quasi-uniform and variable-spacing nodes on domains whose boundaries are represented as computer-aided design (CAD) models, more specifically non-uniform rational B-splines (NURBS). This new algorithm enables the solution of partial differential equations within the volumes enclosed by these CAD models using (collocation-based) meshless numerical discretizations. Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary, then uses the NURBS representation in conjunction with the surface nodes to generate nodes within the volume enclosed by the NURBS surface. We provide evidence for the quality of these node sets by analyzing them in terms of local regularity and separation distances. Finally, we demonstrate that these node sets are well-suited (both in terms of accuracy and numerical stability) for meshless radial basis function generated finite differences discretizations of the Poisson, Navier-Cauchy, and heat equations. Our algorithm constitutes an important step in bridging the field of node generation for meshless discretizations with isogeometric analysis.