We propose a modified local discontinuous Galerkin (LDG) method for second-order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete ...
详细信息
ISBN:
(纸本)9783319399294;9783319399270
We propose a modified local discontinuous Galerkin (LDG) method for second-order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincare-Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes.
We present a comparative study of integral operators used in nonlocal problems. The size of nonlocality is determined by the parameter delta. The authors recently discovered a way to incorporate local boundary conditi...
详细信息
ISBN:
(纸本)9783319399294;9783319399270
We present a comparative study of integral operators used in nonlocal problems. The size of nonlocality is determined by the parameter delta. The authors recently discovered a way to incorporate local boundary conditions into nonlocal problems. We construct two nonlocal operators which satisfy local homogeneous Neumann boundary conditions. We compare the bulk and boundary behaviors of these two to the operator that enforces nonlocal boundary conditions. We construct approximations to each operator using perturbation expansions in the form of Taylor polynomials by consistently keeping the size of expansion neighborhood equal to delta. In the bulk, we show that one of these two operators exhibits similar behavior with the operator that enforces nonlocal boundary conditions.
Recently, we showed in (O. Kolb, SIAM J. Numer. Anal., 52 (2014), pp. 2335-2355) for which parameter range the compact third order WENO reconstruction procedure introduced in (D. Levy, G. Puppo, and G. Russo, SIAM J. ...
详细信息
ISBN:
(纸本)9783319399294;9783319399270
Recently, we showed in (O. Kolb, SIAM J. Numer. Anal., 52 (2014), pp. 2335-2355) for which parameter range the compact third order WENO reconstruction procedure introduced in (D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 22 (2000), pp. 656-672) reaches the optimal order of accuracy (h(3) in the smooth case and h(2) near discontinuities). This is the case for the parameter choice epsilon = Kh(q) in the weight design with q <= 3 and pq >= 2, where p >= 1 is the exponent used in the computation of the weights in theWENO scheme. While these theoretical results for the convergence rates of theWENO reconstruction procedure could also be validated in the numerical tests, the application within the semi-discrete central scheme of (A. Kurganov, and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461-1488) together with a third order TVD-Runge-Kutta scheme for the time integration did not yield a third order accurate scheme in total for q > 2. The aim of this follow-up paper is to explain this observation with further analytical and numerical results.
This paper reviews the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type w...
详细信息
ISBN:
(纸本)9783319416403;9783319416380
This paper reviews the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L-2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.
In this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation p...
详细信息
ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
In this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation points, and supporting quadrature and interpolation on unbounded sets. We also consider several profit indicators that are suitable to drive the adaptation process. We then use such algorithm to solve an important test case in Uncertainty Quantification problem, namely the Darcy equation with lognormal permeability random field, and compare the results with those obtained with the quasi-optimal sparse grids based on profit estimates, which we have proposed in our previous works (cf. e. g. Convergence of quasi-optimal sparse grids approximation of Hilbert-valued functions: application to random elliptic PDEs). To treat the case of rough permeability fields, in which a sparse grid approach may not be suitable, we propose to use the adaptive sparse grid quadrature as a control variate in a Monte Carlo simulation. Numerical results show that the adaptive sparse grids have performances similar to those of the quasi-optimal sparse grids and are very effective in the case of smooth permeability fields. Moreover, their use as control variate in a Monte Carlo simulation allows to tackle efficiently also problems with rough coefficients, significantly improving the performances of a standard Monte Carlo scheme.
In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time fin...
详细信息
ISBN:
(纸本)9783319399294;9783319399270
In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time finite element methods based on a discontinuous Galerkin approximation of the time variable are used. The spatial approximation of the flow problem is done by mixed finite element methods. The stability of the approach is illustrated by numerical experiments. The presented variational space-time framework is of higher order accuracy such that problems with high fluctuations become feasible. Moreover, it offers promising potential for the simulation of the fully dynamic Biot-Allard system coupling an elastic wave equation for solid's deformation with single-phase flow for fluid infiltration.
暂无评论