C. Beck, F. Hornung, Martin Hutzenthaler, Arnulf Jentzen, T. Kruse
Abstract One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.
{"title":"Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations","authors":"C. Beck, F. Hornung, Martin Hutzenthaler, Arnulf Jentzen, T. Kruse","doi":"10.1515/jnma-2019-0074","DOIUrl":"https://doi.org/10.1515/jnma-2019-0074","url":null,"abstract":"Abstract One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78439220","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}
Abstract In [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂. A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented.
在[Nkemzi and Jung, 2013]中，给出了计算轴对称边缘拉普拉斯函数边缘通量强度函数的显式提取公式。本文提出了一种新的傅里叶-有限元方法的改进，用于有效地数值处理轴对称域上泊松方程的边值问题Ω∧有边的∈3。该方法的新颖之处在于利用边缘通量强度函数的显式提取公式，定义了二维子午线Ω²上的简化边值问题的有限元解的后处理过程。先验误差估计表明，后处理有限元策略在规则网格上具有最优的收敛速度。数值实验验证了理论结果。
{"title":"The Fourier-finite-element method for Poisson’s equation in three-dimensional axisymmetric domains with edges: Computing the edge flux intensity functions","authors":"B. Nkemzi, M. Jung","doi":"10.1515/jnma-2019-0002","DOIUrl":"https://doi.org/10.1515/jnma-2019-0002","url":null,"abstract":"Abstract In [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂. A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85094661","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}
Pub Date : 2019-06-26DOI: 10.1515/jnma-2019-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/jnma-2019-frontmatter2","DOIUrl":"https://doi.org/10.1515/jnma-2019-frontmatter2","url":null,"abstract":"","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74228866","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}
A. Sokolov, O. Davydov, D. Kuzmin, Alexander Westermann, S. Turek
Abstract In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable.
{"title":"A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds","authors":"A. Sokolov, O. Davydov, D. Kuzmin, Alexander Westermann, S. Turek","doi":"10.1515/jnma-2018-0097","DOIUrl":"https://doi.org/10.1515/jnma-2018-0097","url":null,"abstract":"Abstract In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76028280","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}
D. Conte, R. D'Ambrosio, M. Moccaldi, B. Paternoster
Abstract In this paper, we present a general class of exponentially fitted two-step peer methods for the numerical integration of ordinary differential equations. The numerical scheme is constructed in order to exploit a-priori known information about the qualitative behaviour of the solution by adapting peer methods already known in literature. Examples of methods with 2 and 3 stages are provided. The effectiveness of this problem-oriented approach is shown through some numerical tests on well-known problems.
{"title":"Adapted explicit two-step peer methods","authors":"D. Conte, R. D'Ambrosio, M. Moccaldi, B. Paternoster","doi":"10.1515/jnma-2017-0102","DOIUrl":"https://doi.org/10.1515/jnma-2017-0102","url":null,"abstract":"Abstract In this paper, we present a general class of exponentially fitted two-step peer methods for the numerical integration of ordinary differential equations. The numerical scheme is constructed in order to exploit a-priori known information about the qualitative behaviour of the solution by adapting peer methods already known in literature. Examples of methods with 2 and 3 stages are provided. The effectiveness of this problem-oriented approach is shown through some numerical tests on well-known problems.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91176176","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}
Abstract The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples.
{"title":"Dual weighted residual error estimation for the finite cell method","authors":"P. Stolfo, A. Rademacher, A. Schröder","doi":"10.1515/jnma-2017-0103","DOIUrl":"https://doi.org/10.1515/jnma-2017-0103","url":null,"abstract":"Abstract The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86894857","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}
Abstract We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.
{"title":"Reduced basis approximations of the solutions to spectral fractional diffusion problems","authors":"A. Bonito, D. Guignard, Ashley R. Zhang","doi":"10.1515/jnma-2019-0053","DOIUrl":"https://doi.org/10.1515/jnma-2019-0053","url":null,"abstract":"Abstract We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82533071","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}
Abstract We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛–sf to fractional PDEs involving elliptic operators. Reinterpreting 𝓛–s as an interpolation operator allows us to write u(s) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation L of the operator whose inverse is projected to the s-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously. A second algorithm is presented to avoid inversion of L. Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.
{"title":"A reduced basis method for fractional diffusion operators II","authors":"Tobias Danczul, J. Schöberl","doi":"10.1515/jnma-2020-0042","DOIUrl":"https://doi.org/10.1515/jnma-2020-0042","url":null,"abstract":"Abstract We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛–sf to fractional PDEs involving elliptic operators. Reinterpreting 𝓛–s as an interpolation operator allows us to write u(s) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation L of the operator whose inverse is projected to the s-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously. A second algorithm is presented to avoid inversion of L. Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83111477","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}
Abstract Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency. The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners. It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.
{"title":"Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems","authors":"O. Axelsson, D. Lukáš","doi":"10.1515/jnma-2017-0064","DOIUrl":"https://doi.org/10.1515/jnma-2017-0064","url":null,"abstract":"Abstract Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency. The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners. It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90064544","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}
Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.
{"title":"Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems","authors":"S. Russell, M. Stynes","doi":"10.1515/jnma-2017-0079","DOIUrl":"https://doi.org/10.1515/jnma-2017-0079","url":null,"abstract":"Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2019-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80034721","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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