Pub Date : 2023-10-19DOI: 10.1142/s0218202523500549
Zhaolong Han, Xiaochuan Tian
This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains.
{"title":"Nonlocal half-ball vector operators on bounded domains: Poincare inequality and its applications","authors":"Zhaolong Han, Xiaochuan Tian","doi":"10.1142/s0218202523500549","DOIUrl":"https://doi.org/10.1142/s0218202523500549","url":null,"abstract":"This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-18DOI: 10.1142/s0218202523500586
Yvon Maday, Carlo Marcati
We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined [Formula: see text] discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined [Formula: see text] space, the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.
{"title":"Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems","authors":"Yvon Maday, Carlo Marcati","doi":"10.1142/s0218202523500586","DOIUrl":"https://doi.org/10.1142/s0218202523500586","url":null,"abstract":"We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined [Formula: see text] discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined [Formula: see text] space, the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"46 7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-04DOI: 10.1142/s0218202523500525
Luca Heltai, Paolo Zunino
Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the infsup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.
{"title":"Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains","authors":"Luca Heltai, Paolo Zunino","doi":"10.1142/s0218202523500525","DOIUrl":"https://doi.org/10.1142/s0218202523500525","url":null,"abstract":"Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the infsup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"40 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":"135547277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1142/s0218202523500550
Tim Laux, Kerrek Stinson
We propose a weak solution theory for the sharp interface limit of the Cahn–Hilliard reaction model, a variational PDE for lithium-ion batteries. An essential feature of this model is the use of Butler–Volmer kinetics for lithium-ion insertion, which arises as a Robin-type boundary condition relating the flux of the chemical potential to the reaction rate, itself a nonlinear function of the chemical potential and the ion concentration. To pass through the nonlinearity as the interface width vanishes, we introduce solution concepts at the diffuse and sharp interface level describing dynamics principally in terms of an optimal dissipation inequality. Using this functional framework and under an energy convergence hypothesis, we show that solutions of the Cahn–Hilliard reaction model converge to a Mullins–Sekerka type geometric evolution equation as the width of the transition layer vanishes.
{"title":"Sharp Interface Limit of the Cahn-Hilliard Reaction Model for Lithium-ion Batteries","authors":"Tim Laux, Kerrek Stinson","doi":"10.1142/s0218202523500550","DOIUrl":"https://doi.org/10.1142/s0218202523500550","url":null,"abstract":"We propose a weak solution theory for the sharp interface limit of the Cahn–Hilliard reaction model, a variational PDE for lithium-ion batteries. An essential feature of this model is the use of Butler–Volmer kinetics for lithium-ion insertion, which arises as a Robin-type boundary condition relating the flux of the chemical potential to the reaction rate, itself a nonlinear function of the chemical potential and the ion concentration. To pass through the nonlinearity as the interface width vanishes, we introduce solution concepts at the diffuse and sharp interface level describing dynamics principally in terms of an optimal dissipation inequality. Using this functional framework and under an energy convergence hypothesis, we show that solutions of the Cahn–Hilliard reaction model converge to a Mullins–Sekerka type geometric evolution equation as the width of the transition layer vanishes.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136271947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1142/s0218202523500562
A. Brunk, H. Egger, O. Habrich, M. Lukacova-Medvid'ova
We propose and analyze a structure-preserving space-time variational discretization method for the Cahn–Hilliard–Navier–Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration-dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.
{"title":"A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn-Hilliard Navier-Stokes system","authors":"A. Brunk, H. Egger, O. Habrich, M. Lukacova-Medvid'ova","doi":"10.1142/s0218202523500562","DOIUrl":"https://doi.org/10.1142/s0218202523500562","url":null,"abstract":"We propose and analyze a structure-preserving space-time variational discretization method for the Cahn–Hilliard–Navier–Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration-dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136272097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1142/s0218202523500537
Guo-Dong Zhang, Xiaofeng Yang
In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the [Formula: see text]-error bound and the [Formula: see text]-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of [Formula: see text] instead of [Formula: see text] for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.
{"title":"Error estimates with low-order polynomial dependence for the fully-discrete finite element Invariant Energy Quadratization scheme of the Allen-Cahn Equation","authors":"Guo-Dong Zhang, Xiaofeng Yang","doi":"10.1142/s0218202523500537","DOIUrl":"https://doi.org/10.1142/s0218202523500537","url":null,"abstract":"In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the [Formula: see text]-error bound and the [Formula: see text]-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of [Formula: see text] instead of [Formula: see text] for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136272102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1142/s0218202524500064
Michal Bathory, Miroslav Bul'ivcek, J. M'alek
We prove that there exists a~large-data and global-in-time weak solution to a~system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a~mechanically and thermally isolated container of any dimension. To overcome the~principle difficulties connected with ill-posedness of the~diffusive Oldroyd-B model in three dimensions, we assume that the~fluid admits a~strengthened dissipation mechanism, at least for excessive elastic deformations. All the~relevant material coefficients are allowed to depend continuously on the~temperature, whose evolution is captured by a~thermodynamically consistent equation. In fact, the~studied model is derived from scratch using only the~balance equations for linear momentum and energy, the~formulation of the~second law of thermodynamics and the~constitutive equation for the~internal energy. The~latter is assumed to be a~linear function of temperature, which simplifies the~model. The~concept of our weak solution incorporates both the~temperature and entropy inequalities, and also the~local balance of total energy provided that the~pressure function exists.
{"title":"Coupling the Navier–Stokes–Fourier equations with the Johnson–Segalman stress-diffusive viscoelastic model: global-in-time and large-data analysis","authors":"Michal Bathory, Miroslav Bul'ivcek, J. M'alek","doi":"10.1142/s0218202524500064","DOIUrl":"https://doi.org/10.1142/s0218202524500064","url":null,"abstract":"We prove that there exists a~large-data and global-in-time weak solution to a~system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a~mechanically and thermally isolated container of any dimension. To overcome the~principle difficulties connected with ill-posedness of the~diffusive Oldroyd-B model in three dimensions, we assume that the~fluid admits a~strengthened dissipation mechanism, at least for excessive elastic deformations. All the~relevant material coefficients are allowed to depend continuously on the~temperature, whose evolution is captured by a~thermodynamically consistent equation. In fact, the~studied model is derived from scratch using only the~balance equations for linear momentum and energy, the~formulation of the~second law of thermodynamics and the~constitutive equation for the~internal energy. The~latter is assumed to be a~linear function of temperature, which simplifies the~model. The~concept of our weak solution incorporates both the~temperature and entropy inequalities, and also the~local balance of total energy provided that the~pressure function exists.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139351392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-02DOI: 10.1142/s0218202523500495
Monica Montardini, Giancarlo Sangalli, Rainer Schneckenleitner, Stefan Takacs, Mattia Tani
We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows for the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory for two-dimensional problems that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines. Additionally, we present an extension of the solver to three-dimensional problems and provide numerical experiments assessing good performance also in that setting.
{"title":"A IETI-DP method for discontinuous Galerkin discretizations in isogeometric analysis with inexact local solvers","authors":"Monica Montardini, Giancarlo Sangalli, Rainer Schneckenleitner, Stefan Takacs, Mattia Tani","doi":"10.1142/s0218202523500495","DOIUrl":"https://doi.org/10.1142/s0218202523500495","url":null,"abstract":"We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows for the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory for two-dimensional problems that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines. Additionally, we present an extension of the solver to three-dimensional problems and provide numerical experiments assessing good performance also in that setting.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"132 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136384633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-31DOI: 10.1142/s0218202523500434
Cesare Bracco, Carlotta Giannelli, Mario Kapl, Rafael Vazquez
Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of [Formula: see text] isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by [Formula: see text] splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.
{"title":"Adaptive isogeometric methods with C1 (truncated) hierarchical splines on planar multi-patch domains","authors":"Cesare Bracco, Carlotta Giannelli, Mario Kapl, Rafael Vazquez","doi":"10.1142/s0218202523500434","DOIUrl":"https://doi.org/10.1142/s0218202523500434","url":null,"abstract":"Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of [Formula: see text] isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by [Formula: see text] splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135194798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-29DOI: 10.1142/s0218202523500422
Jarle Sogn, Stefan Takacs
We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch’s framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble’s framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.
{"title":"Multigrid solvers for isogeometric discretizations of the second biharmonic problem","authors":"Jarle Sogn, Stefan Takacs","doi":"10.1142/s0218202523500422","DOIUrl":"https://doi.org/10.1142/s0218202523500422","url":null,"abstract":"We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch’s framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble’s framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135740873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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