Adaptive higher-order finite element methods(hp-FEM)are well known for their potential of exceptionally fast(exponential)convergence.However,most hp-FEM codes remain in an academic setting due to an extreme algorithmi...Adaptive higher-order finite element methods(hp-FEM)are well known for their potential of exceptionally fast(exponential)convergence.However,most hp-FEM codes remain in an academic setting due to an extreme algorithmic complexity of hp-adaptivity algorithms.This paper aims at simplifying hpadaptivity for H(curl)-conforming approximations by presenting a novel technique of arbitrary-level hanging nodes.The technique is described and it is demonstrated numerically that it makes adaptive hp-FEM more efficient compared to hp-FEM on regular meshes and meshes with one-level hanging nodes.展开更多
We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust ...We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method （FEM） on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.展开更多
We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary le...We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary lengths and polynomial degrees(hp-FEM).We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.展开更多
We are concerned with a model of ionic polymer-metal composite(IPMC)materials that consists of a coupled system of the Poisson and Nernst-Planck equations,discretized by means of the finite element method(FEM).We show...We are concerned with a model of ionic polymer-metal composite(IPMC)materials that consists of a coupled system of the Poisson and Nernst-Planck equations,discretized by means of the finite element method(FEM).We show that due to the transient character of the problem it is efficient to use adaptive algorithms that are capable of changing the mesh dynamically in time.We also show that due to large qualitative and quantitative differences between the two solution components,it is efficient to approximate them on different meshes using a novel adaptive multimesh hp-FEM.The study is accompanied with numerous computations and comparisons of the adaptive multimesh hp-FEMwith several other adaptive FEM algorithms.展开更多
We present a novel compression algorithm for 2D scientific data and images based on exponentially-convergent adaptive higher-order finite element methods(FEM).So far,FEM has been used mainly for the solution of part...We present a novel compression algorithm for 2D scientific data and images based on exponentially-convergent adaptive higher-order finite element methods(FEM).So far,FEM has been used mainly for the solution of partial differential equations(PDE),but we show that it can be applied to data and image compression easily.The adaptive compression algorithm is trivial compared to adaptive FEM algorithms for PDE since the error estimation step is not present.The method attains extremely high compression rates and is able to compress a data set or an image with any prescribed error tolerance.Compressed data and images are stored in the standard FEM format,which makes it possible to analyze them using standard PDE visualization software.Numerical examples are shown.The method is presented in such a way that it can be understood by readers who may not be experts of the finite element method.展开更多
文摘Adaptive higher-order finite element methods(hp-FEM)are well known for their potential of exceptionally fast(exponential)convergence.However,most hp-FEM codes remain in an academic setting due to an extreme algorithmic complexity of hp-adaptivity algorithms.This paper aims at simplifying hpadaptivity for H(curl)-conforming approximations by presenting a novel technique of arbitrary-level hanging nodes.The technique is described and it is demonstrated numerically that it makes adaptive hp-FEM more efficient compared to hp-FEM on regular meshes and meshes with one-level hanging nodes.
文摘We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method （FEM） on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.
基金the support of the Czech Science Foundation,proj-ects No.102/07/0496 and 102/05/0629the Grant Agency of the Academy of Sciences of the Czech Republic,project No.IAA100760702the Academy of Sciences of the Czech Republic,Institutional Research Plan No.AV0Z10190503。
文摘We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary lengths and polynomial degrees(hp-FEM).We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.
基金supported by the Grant Agency of the Academy of Sciences of the Czech Republic under Grant No.IAA100760702and by the U.S.Department of Energy Research Subcontract No.00089911+1 种基金The third author acknowledges the financial support of the U.S.Office of Naval Research under Award N000140910218The fourth author acknowledges the financial support of the Estonian Ministry of Education,grant#SF0180008s08.
文摘We are concerned with a model of ionic polymer-metal composite(IPMC)materials that consists of a coupled system of the Poisson and Nernst-Planck equations,discretized by means of the finite element method(FEM).We show that due to the transient character of the problem it is efficient to use adaptive algorithms that are capable of changing the mesh dynamically in time.We also show that due to large qualitative and quantitative differences between the two solution components,it is efficient to approximate them on different meshes using a novel adaptive multimesh hp-FEM.The study is accompanied with numerous computations and comparisons of the adaptive multimesh hp-FEMwith several other adaptive FEM algorithms.
基金the financial support of the Czech Science Foundation(Project No.102/07/0496)and of the Grant Agency of the Academy of Sciences of the Czech Republic(Project No.IAA100760702)。
文摘We present a novel compression algorithm for 2D scientific data and images based on exponentially-convergent adaptive higher-order finite element methods(FEM).So far,FEM has been used mainly for the solution of partial differential equations(PDE),but we show that it can be applied to data and image compression easily.The adaptive compression algorithm is trivial compared to adaptive FEM algorithms for PDE since the error estimation step is not present.The method attains extremely high compression rates and is able to compress a data set or an image with any prescribed error tolerance.Compressed data and images are stored in the standard FEM format,which makes it possible to analyze them using standard PDE visualization software.Numerical examples are shown.The method is presented in such a way that it can be understood by readers who may not be experts of the finite element method.
