In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $varDelta tle O(h^{2})$. We show that we only need the lower bound $varDelta tge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $varDelta tle C$ (for globally Lipschitz) or $varDelta tle O(h^{gamma })$ (for locally Lipschitz) for some constant $C>0$ arising from the PDE, or some $gamma < 2$, respectively. In most situations in practical models, the latter condition becomes $varDelta t le O( h^{2/3} )$ in 2D and $varDelta t le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.
在本文中，我们扩展了之前在 Faragó, I., Karátson, J. 和 Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems.IMA J. Numer.Anal.，32，1541-1573）关于非线性抛物 PDE 系统的离散最大最小原理（DMP）。我们提出了一种线性隐式方案，只需解决时间层上的线性问题。我们得到的 DMP 没有限制性条件 $varDelta tle O(h^{2})$。我们证明，我们只需要下界 $varDelta tge O(h^{2})$，此外，根据给定非线性的 Lipschitz 条件，对于由 PDE 产生的某个常数 $C>0$，或者某个 $gamma < 2$，上界仅仅是 $varDelta tle C$（对于全局 Lipschitz）或 $varDelta tle O(h^{gamma})$（对于局部 Lipschitz）。在实际模型的大多数情况下，后一个条件在二维模型中变成 $varDelta t le O( h^{2/3} )$ ，在三维模型中变成 $varDelta t le O( h )$ 。此外，还介绍了各种现实生活中的例子，在这些例子中，可以应用这些结果来获得与物理相关的数值解。
{"title":"Discrete maximum-minimum principle for a linearly implicit scheme for nonlinear parabolic FEM problems under weakened time restrictions","authors":"István Faragó, Róbert Horváth, János Karátson","doi":"10.1093/imanum/drae072","DOIUrl":"https://doi.org/10.1093/imanum/drae072","url":null,"abstract":"In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $varDelta tle O(h^{2})$. We show that we only need the lower bound $varDelta tge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $varDelta tle C$ (for globally Lipschitz) or $varDelta tle O(h^{gamma })$ (for locally Lipschitz) for some constant $C&gt;0$ arising from the PDE, or some $gamma &lt; 2$, respectively. In most situations in practical models, the latter condition becomes $varDelta t le O( h^{2/3} )$ in 2D and $varDelta t le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0<r_{k}:=tau _{k}/tau _{k-1}< r_{max }approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.
{"title":"An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach","authors":"Yifan Wei, Jiwei Zhang, Chengchao Zhao, Yanmin Zhao","doi":"10.1093/imanum/drae057","DOIUrl":"https://doi.org/10.1093/imanum/drae057","url":null,"abstract":"An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0&lt;r_{k}:=tau _{k}/tau _{k-1}&lt; r_{max }approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142321471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.
{"title":"A higher order multiscale method for the wave equation","authors":"Felix Krumbiegel, Roland Maier","doi":"10.1093/imanum/drae059","DOIUrl":"https://doi.org/10.1093/imanum/drae059","url":null,"abstract":"In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142275655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.
{"title":"High-order energy stable discrete variational derivative schemes for gradient flows","authors":"Jizu Huang","doi":"10.1093/imanum/drae062","DOIUrl":"https://doi.org/10.1093/imanum/drae062","url":null,"abstract":"The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"199 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142275656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.
{"title":"A mesh-independent method for second-order potential mean field games","authors":"Kang Liu, Laurent Pfeiffer","doi":"10.1093/imanum/drae061","DOIUrl":"https://doi.org/10.1093/imanum/drae061","url":null,"abstract":"This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz
A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(mathbb{T}^{2})$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $tau ^{s/2}$ in $L^{2}(mathbb{T}^{2})$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.
{"title":"Low regularity error estimates for the time integration of 2D NLS","authors":"Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz","doi":"10.1093/imanum/drae054","DOIUrl":"https://doi.org/10.1093/imanum/drae054","url":null,"abstract":"A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(mathbb{T}^{2})$ with $s&gt;0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s&gt;1$ is overcome. Rates of convergence of order $tau ^{s/2}$ in $L^{2}(mathbb{T}^{2})$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142231539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.
{"title":"A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes","authors":"Haifeng Ji, Dong Liang, Qian Zhang","doi":"10.1093/imanum/drae053","DOIUrl":"https://doi.org/10.1093/imanum/drae053","url":null,"abstract":"This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier
In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška–Brezzi theory in Banach spaces, allows to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.
{"title":"New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations","authors":"Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier","doi":"10.1093/imanum/drae052","DOIUrl":"https://doi.org/10.1093/imanum/drae052","url":null,"abstract":"In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška–Brezzi theory in Banach spaces, allows to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"51 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142142448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.
{"title":"Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes","authors":"Sören Bartels, Philipp Tscherner","doi":"10.1093/imanum/drae050","DOIUrl":"https://doi.org/10.1093/imanum/drae050","url":null,"abstract":"It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.
{"title":"Computing Klein-Gordon Spectra","authors":"Frank Rösler, Christiane Tretter","doi":"10.1093/imanum/drae032","DOIUrl":"https://doi.org/10.1093/imanum/drae032","url":null,"abstract":"We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"98 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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