The Experts below are selected from a list of 9609 Experts worldwide ranked by ideXlab platform
Shangyou Zhang - One of the best experts on this subject based on the ideXlab platform.
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Erratum to “A C1-P2 finite element without nodal basis”
2021Co-Authors: Shangyou ZhangAbstract:A new Interpolation Operator is defined, which preserves only P2 polynomials locally
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a c 0 weak galerkin finite element method for the biharmonic equation
2014Co-Authors: Junping Wang, Shangyou ZhangAbstract:A \(C^0\)-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for \(C^0\) functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete \(H^2\) norm and the standard \(H^1\) and \(L^2\) norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang Interpolation Operator is constructed to assist the corresponding error estimates. This refined Interpolation preserves the volume mass of order \((k+1-d)\) and the surface mass of order \((k+2-d)\) for the \(P_{k+2}\) finite element functions in \(d\)-dimensional space.
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a c 0 weak galerkin finite element method for the biharmonic equation
2012Co-Authors: Junping Wang, Shangyou ZhangAbstract:A C^0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C^0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H^2 norm and the L^2 norm, for the weak Galerkin finite element solution. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang Interpolation Operator is constructed to assist the corresponding error estimate. This refined Interpolation preserves the volume mass of order (k+1-d) and the surface mass of order (k+2-d) for the P_{k+2} finite element functions in d-dimensional space.
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a new divergence free Interpolation Operator with applications to the darcy stokes brinkman equations
2010Co-Authors: Shangyou ZhangAbstract:A new local Interpolation Operator, preserving the divergence, is constructed explicitly for the Hsieh-Clough-Tocher divergence-free element. A divergence-free finite element method is applied to the Darcy-Stokes-Brinkman flow in a mixed region of both free and porous media. The method is of optimal order as well for the Darcy flow as for the Stokes flow (which is not the case for most other finite elements). Compared to the existing nonconforming elements, the divergence-free element method provides a continuous solution for the velocity which is also an orthogonal projection within a Hilbert subspace of the true velocity. Numerical tests supporting the theory are presented.
Edriss S. Titi - One of the best experts on this subject based on the ideXlab platform.
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continuous data assimilation for the three dimensional navier stokes α model
2016Co-Authors: Debora A F Albanez, Edriss S. Titi, Helena Nussenzveig J LopesAbstract:© 2016 - IOS Press and the authors. All rights reserved. Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier-Stokes equations; instead we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier-Stokes-α equations. This algorithm consists of introducing a nudging process through a general type of approximation Interpolation Operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier-Stokes-α equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters.
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Continuous Data Assimilation Using General Interpolant Observables
2014Co-Authors: Abderrahim Azouani, Eric Olson, Edriss S. TitiAbstract:We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier–Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation Interpolation Operator exists. Under the assumption that the observational measurements are free of noise, our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data.
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continuous data assimilation using general interpolant observables
2013Co-Authors: Abderrahim Azouani, Eric Olson, Edriss S. TitiAbstract:We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier--Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation Interpolation Operator exists. Our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data.
Xingqiang Yang - One of the best experts on this subject based on the ideXlab platform.
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constructing c1 triangular patch of degree six by the boolean sum of an approximation Operator and an Interpolation Operator
2011Co-Authors: Caiming Zhang, Xingqiang YangAbstract:A new method to construct C1 triangular patches which satisfy the given boundary curves and cross-boundary slopes is presented. The Boolean sum of an approximation Operator and an Interpolation Operator is employed to construct the triangular patch. The approximation Operator is used to construct a polynomial patch of degree six. The polynomial of degree six affords more freedoms, which makes the approximation Operator not only approximate the given boundary Interpolation conditions but also have a better approximation precision in the interior of the triangle, so that the triangular patch has a better precision on both the boundary and the interior of the triangular domain. The Interpolation Operator is utilized to build an Interpolation patch which satisfies the given boundary conditions. The Boolean sum of the approximation and Interpolation patches forms the triangular patch. Comparison results of the new method with other three methods are given.
Fedotov A. - One of the best experts on this subject based on the ideXlab platform.
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Estimate of the norm of the Lagrange Interpolation Operator in a multidimensional Sobolev space
2020Co-Authors: Fedotov A.Abstract:We obtain an estimate of the norm of the Lagrange Interpolation Operator in a multidimensional Sobolev space. It is shown that, under a suitable choice of the sequence of multi-indices, Interpolation polynomials converge to the interpolated function and their rate of convergence is of the order of the best approximation of this function. © Nauka/Interperiodica 2007
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Estimate of the norm of the Lagrange Interpolation Operator in the multidimensional weighted Sobolev space
2020Co-Authors: Fedotov A.Abstract:© 2016, Pleiades Publishing, Ltd.An estimate of the norm of the Lagrange Interpolation Operator in the multidimensional weighted Sobolev space is obtained. It is shown that, under a certain choice of the sequence of multi-indices, the interpolating polynomials converge to the interpolated function and the rate of convergence is of the order of the best approximation of this function by algebraic polynomials in this space
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Lebesgue constant estimation in multidimensional Sobolev space
2020Co-Authors: Fedotov A.Abstract:The norm estimation of the Lagrange Interpolation Operator is obtained. It is shown that the rate of convergence of the interpolative polynomials depends on the choice of the sequence of multiindices and, for some sequences, is equal to the rate of the best approximation of the interpolated function
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Estimate of the Norm of the Hermite—Fejér Interpolation Operator in Sobolev Spaces
2020Co-Authors: Fedotov A.Abstract:© 2019, Pleiades Publishing, Ltd. Upper bounds for the norms of Hermite—Fejér Interpolation Operators in one-dimensional and multidimensional periodic Sobolev spaces are obtained. It is shown that, in the one-dimensional case, the norm of this Operator is bounded. In the multidimensional case, the upper bound depends on the ratio of the numbers of nodes on separate coordinates
Marcin Skotniczny - One of the best experts on this subject based on the ideXlab platform.
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application of multi agent paradigm to hp adaptive projection based Interpolation Operator
2013Co-Authors: Piotr Gurgul, Marcin Sieniek, Krzysztof Magiera, Marcin SkotnicznyAbstract:Abstract In this paper we discuss applications and design of the agent-based, hp-adaptive projection-based Interpolation (PBI) Operator. We describe the use of mesh adaptation process to produce a faithful representation of an input image in the Finite Element (FE) space. This can be used, in turn, to generate from the input bitmap continuous material functions required for further FE computations. We propose an agent-based architecture suitable for localized implementation of the PBI Operator. Finally, we show how to apply it to an exemplary problem of austenite–ferrite phase transformation.
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agent oriented image processing with the hp adaptive projection based Interpolation Operator
2011Co-Authors: Marcin Sieniek, Piotr Gurgul, Krzysztof Magiera, Marcin Skotniczny, Maciej PaszynskiAbstract:In this paper we discuss applications and design of the agent-oriented, hp-adaptive projection-based Interpolation technique. We describe the use of the mesh adaptation process to produce the most faithful representation of the input image in the Finite Element space. We discuss the advantages of the agent-oriented application model both in general and in terms of the hp-adaptive application properties. Lastly, we describe a sample problem used as a proof of concept.
