The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient compression algorithm. We demonstrate experimentally, that the new algorithm can achieve better compression ratios than standard JPEG compression with no visible loss of quality on many images. The mathematical core of this work shows that Binev’s optimal tree approximation algorithm is applicable to image compression with high probability, when we assume small additive Gaussian noise on the pixels of the image.
{"title":"Adaptive Image Compression via Optimal Mesh Refinement","authors":"Michael Feischl, Hubert Hackl","doi":"10.1515/cmam-2023-0097","DOIUrl":"https://doi.org/10.1515/cmam-2023-0097","url":null,"abstract":"The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient compression algorithm. We demonstrate experimentally, that the new algorithm can achieve better compression ratios than standard JPEG compression with no visible loss of quality on many images. The mathematical core of this work shows that Binev’s optimal tree approximation algorithm is applicable to image compression with high probability, when we assume small additive Gaussian noise on the pixels of the image.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates.
{"title":"A Posteriori Error Estimation for the Optimal Control of Time-Periodic Eddy Current Problems","authors":"Monika Wolfmayr","doi":"10.1515/cmam-2023-0119","DOIUrl":"https://doi.org/10.1515/cmam-2023-0119","url":null,"abstract":"This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"130 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L2L^{2}-norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.
{"title":"Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations","authors":"Kallol Ray, Deepjyoti Goswami, Saumya Bajpai","doi":"10.1515/cmam-2023-0035","DOIUrl":"https://doi.org/10.1515/cmam-2023-0035","url":null,"abstract":"In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0035_ineq_0001.png\" /> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"22 9","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0001.png" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0002.png" /> <jats:tex-math>P_{7}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Bell finite element by restricting its normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0003.png" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0004.png" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, and its second normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0004.png" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0006.png" /> <jats:tex-math>P_{4}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cmam-2023-0068_ineq_0003.png" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial space. We show the method converges at order 7 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlin
{"title":"A 𝐶1-𝑃7 Bell Finite Element on Triangle","authors":"Xuejun Xu, Shangyou Zhang","doi":"10.1515/cmam-2023-0068","DOIUrl":"https://doi.org/10.1515/cmam-2023-0068","url":null,"abstract":"We construct a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0001.png\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0002.png\" /> <jats:tex-math>P_{7}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Bell finite element by restricting its normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0003.png\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0004.png\" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, and its second normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0004.png\" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0006.png\" /> <jats:tex-math>P_{4}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0003.png\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial space. We show the method converges at order 7 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlin","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"128 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p∈L2(Ω){pin L^{2}(Omega)}. The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.
从给定的流速重建压力是一项在各种应用中出现的任务，标准方法使用Navier-Stokes方程来推导压力p的泊松问题。然而，这种方法人为地增加了对解和数据的规则性要求。在这种情况下，我们提出并分析了两种替代技术来确定p∈l2¹(Ω) {p In L^{2}(Omega)}。第一个是一个超弱变分公式，应用分部积分法将所有导数平移到测试函数。给出了合适的有限元离散化方法，并证明了Galerkin-Petrov方法的最佳收敛性。第二种方法是对原始梯度方程采用最小二乘法，将其重新表述并求解为人工Stokes系统。为了简化将给定的速度并入右边，我们在推导中假定速度场是螺线形的。然而，这个假设不是限制性的，因为我们可以使用非散度近似，甚至是可压缩速度。数值实验证实了两种方法的最优先验误差估计。
{"title":"Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation","authors":"Douglas R. Q. Pacheco, Olaf Steinbach","doi":"10.1515/cmam-2021-0242","DOIUrl":"https://doi.org/10.1515/cmam-2021-0242","url":null,"abstract":"Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure <jats:italic>p</jats:italic>. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2021-0242_eq_0228.png\" /> <jats:tex-math>{pin L^{2}(Omega)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"126 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.
{"title":"Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation","authors":"Hans Peter Stimming, Xin Wen, Norbert J. Mauser","doi":"10.1515/cmam-2023-0096","DOIUrl":"https://doi.org/10.1515/cmam-2023-0096","url":null,"abstract":"Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"52 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135161159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.
{"title":"A Conforming Virtual Element Method for Parabolic Integro-Differential Equations","authors":"Sangita Yadav, Meghana Suthar, Sarvesh Kumar","doi":"10.1515/cmam-2023-0061","DOIUrl":"https://doi.org/10.1515/cmam-2023-0061","url":null,"abstract":"Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136059360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Timon S. Gutleb, Norbert J. Mauser, Michele Ruggeri, Hans Peter Stimming
Abstract We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrödinger equation.
{"title":"A Time Splitting Method for the Three-Dimensional Linear Pauli Equation","authors":"Timon S. Gutleb, Norbert J. Mauser, Michele Ruggeri, Hans Peter Stimming","doi":"10.1515/cmam-2023-0094","DOIUrl":"https://doi.org/10.1515/cmam-2023-0094","url":null,"abstract":"Abstract We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrödinger equation.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"100 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.
{"title":"Convergence of the Incremental Projection Method Using Conforming Approximations","authors":"Robert Eymard, David Maltese","doi":"10.1515/cmam-2023-0038","DOIUrl":"https://doi.org/10.1515/cmam-2023-0038","url":null,"abstract":"Abstract We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We construct three H-curl-curl finite elements. The P2 P_{2} and P3 P_{3} vector finite element spaces are both enriched by one common P4 P_{4} bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any P1 P_{1} H-curl-curl conforming finite element, the P1 P_{1} H-curl-curl nonconforming finite element is constructed with three additional P4 P_{4} bubbles. Numerical tests are presented, confirming the conformity and the optimal order of convergence.
{"title":"Three Low Order H-Curl-Curl Finite Elements on Triangular Meshes","authors":"Shangyou Zhang","doi":"10.1515/cmam-2023-0140","DOIUrl":"https://doi.org/10.1515/cmam-2023-0140","url":null,"abstract":"Abstract We construct three H-curl-curl finite elements. The <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> P_{2} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> P_{3} vector finite element spaces are both enriched by one common <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> P_{4} bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> P_{1} H-curl-curl conforming finite element, the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> P_{1} H-curl-curl nonconforming finite element is constructed with three additional <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> P_{4} bubbles. Numerical tests are presented, confirming the conformity and the optimal order of convergence.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"178 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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