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In this paper we study two different ways of coupling a local operator with a nonlocal one so that the resulting equation is related to an energy functional. In the first strategy the coupling is given via source terms in the equa...
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In this paper we study two different ways of coupling a local operator with a nonlocal one so that the resulting equation is related to an energy functional. In the first strategy the coupling is given via source terms in the equation, and in the second one a flux condition in the local part appears. For both models we prove existence and uniqueness of a solution that is obtained via direct minimization of the related energy functional. In the second part of this paper we extend these ideas to local/nonlocal elasticity models in which we couple classical local elasticity with nonlocal peridynamics.
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In a paper by R. Duran, A. Lombardi, and the authors (2007) the finite element method was applied to a non-homogeneous Neumann problem on a cuspidal domain Ω ∩ R~2, and quasi-optimal order error estimates in the energy norm were...
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In a paper by R. Duran, A. Lombardi, and the authors (2007) the finite element method was applied to a non-homogeneous Neumann problem on a cuspidal domain Ω ∩ R~2, and quasi-optimal order error estimates in the energy norm were obtained for certain graded meshes. In this paper, we study the error in the L~2 norm obtaining similar results by using graded meshes of the type considered in that paper. Since many classical results in the theory Sobolev spaces do not apply to the domain under consideration, our estimates require a particular duality treatment working on appropriate weighted spaces. On the other hand, since the discrete domain Ω_h verifies Ω ∩ Ω_h, in the above-mentioned paper the source term of the Poisson problem was taken equal to 0 outside Ω in the variational discrete formulation. In this article we also consider the case in which this condition does not hold and obtain more general estimates, which can be useful in different problems, for instance in the study of the effect of numerical integration, or in eigenvalue approximations.
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A finite element scheme for an entirely fractional Allen–Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian...
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A finite element scheme for an entirely fractional Allen–Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing fractional parameter and usual derivative in time, is discussed within the framework of the Γ-convergence theory.
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We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensiona...
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We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra. Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree k ≥ 0, we prove error estimates of order j+1 when the vector field being approximated has components in W~(j+1,P), for triangles or tetrahedra, where 0 ≤ j ≤ k and 1 ≤ p ≤ ∞. These results are new even in the twodimensional case. Indeed, the estimate was known only in the case j = k. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for k = 0.
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摘要 :
We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensiona...
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We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra. Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree k ≥ 0, we prove error estimates of order j+1 when the vector field being approximated has components in W~(j+1,P), for triangles or tetrahedra, where 0 ≤ j ≤ k and 1 ≤ p ≤ ∞. These results are new even in the twodimensional case. Indeed, the estimate was known only in the case j = k. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for k = 0.
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摘要 :Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this wor...
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Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.
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In this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous Neumann problem in a cuspidal domain. Since the domain is not Lipschitz, many of the results on Sobolev spaces, which are ...
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In this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous Neumann problem in a cuspidal domain. Since the domain is not Lipschitz, many of the results on Sobolev spaces, which are fundamental in the usual error analysis, do not apply. Therefore, we need to work with weighted Sobolev spaces and to develop some new theorems on traces and extensions. We show that, in the domain considered here, suboptimal order can be obtained with quasi-uniform meshes even when the exact solution is in H~2, and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes.
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An average interpolation is introduced for 3-rectangles and tetrahedra, and optimal order error estimates in the H~1 norm are proved. The constant in the estimate depends "weakly" (improving the results given in Duran (Math. Comp....
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An average interpolation is introduced for 3-rectangles and tetrahedra, and optimal order error estimates in the H~1 norm are proved. The constant in the estimate depends "weakly" (improving the results given in Duran (Math. Comp. 68 (1999) 187-199) on the uniformity of the mesh in each direction. For tetrahedra, the constant also depends on the maximum angle of the element. On the other hand, merging several known results (Acosta and Duran, SIAM J. Numer. Anal. 37 (1999) 18-36; Duran, Math. Comp. 68 (1999) 187-199; Krizek, SIAM J. Numer. Anal. 29 (1992) 513-520; Al Shenk, Math. Comp. 63 (1994) 105-119), we prove optimal order error for the P_1-Lagrange interpolation in W~(1,p), p > 2, with a constant depending on p as well as the maximum angle of the element. Again, under the maximum angle condition, optimal order error estimates are obtained in the H~1 norm for higher degree interpolations.
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We Study the behavior of positive solutions of the system. in a bounded domain with the boundary conditions. and the initial data (u0,v0). We find conditions on the functions a,b,f,g,r,s that guarantee the global existence (or fin...
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We Study the behavior of positive solutions of the system. in a bounded domain with the boundary conditions. and the initial data (u0,v0). We find conditions on the functions a,b,f,g,r,s that guarantee the global existence (or finite time blow-up) of positive solutions for every (u0,v0).
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