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...
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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.
We propose the construction of a class of L-2 stable quasi-interpolation operators onto the space of splines on tensor-product meshes, in any space dimension. The estimate we propose is robust with respect to knot rep...
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ISBN:
(纸本)9783319416403;9783319416380
We propose the construction of a class of L-2 stable quasi-interpolation operators onto the space of splines on tensor-product meshes, in any space dimension. The estimate we propose is robust with respect to knot repetition and to knot "vicinity" ( up to p + 1 knots), so it applies to the most general scenario in which the B-spline functions are known to be well defined.
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...
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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...
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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 the present study, we establish two new block variants of the Conjugate Orthogonal Conjugate Gradient (COCG) and the Conjugate A-Orthogonal Conjugate Residual (COCR) Krylov subspace methods for solving complex symm...
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ISBN:
(纸本)9783319399294;9783319399270
In the present study, we establish two new block variants of the Conjugate Orthogonal Conjugate Gradient (COCG) and the Conjugate A-Orthogonal Conjugate Residual (COCR) Krylov subspace methods for solving complex symmetric linear systems with multiple right hand sides. The proposed Block iterative solvers can fully exploit the complex symmetry property of coefficient matrix of the linear system. We report on extensive numerical experiments to show the favourable convergence properties of our newly developed Block algorithms for solving realistic electromagnetic simulations.
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 ...
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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 introduce an abstract concept for decomposing spaces with respect to a substructuring of a bounded domain. In this setting we define weakly conforming finite element approximations of quadratic minimization problem...
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ISBN:
(纸本)9783319399294;9783319399270
We introduce an abstract concept for decomposing spaces with respect to a substructuring of a bounded domain. In this setting we define weakly conforming finite element approximations of quadratic minimization problems. Within a saddle point approach the reduction to symmetric positive Schur complement systems on the skeleton is analyzed. Applications include weakly conforming variants of least squares and minimal residuals.
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