Numerical Methods for Partial Differential Equations最新文献

{"title":"New quadratic/serendipity finite volume element solutions on arbitrary triangular/quadrilateral meshes","authors":"Yanhui Zhou","doi":"10.1002/num.23093","DOIUrl":"https://doi.org/10.1002/num.23093","url":null,"abstract":"By postprocessing quadratic and eight‐node serendipity finite element solutions on arbitrary triangular and quadrilateral meshes, we obtain new quadratic/serendipity finite volume element solutions for solving anisotropic diffusion equations. The postprocessing procedure is implemented in each element independently, and we only need to solve a 6‐by‐6 (resp. 8‐by‐8) local linear algebraic system for triangular (resp. quadrilateral) element. The novelty of this paper is that, by designing some new quadratic dual meshes, and adding six/eight special constructed element‐wise bubble functions to quadratic/serendipity finite element solutions, we prove that the postprocessed solutions satisfy local conservation property on the new dual meshes. In particular, for any full anisotropic diffusion tensor, arbitrary triangular and quadrilateral meshes, we present a general framework to prove the existence and uniqueness of new quadratic/serendipity finite volume element solutions, which is better than some existing ones. That is, the existing theoretical results are improved, especially we extend the traditional rectangular assumption to arbitrary convex quadrilateral mesh. As a byproduct, we also prove that the new solutions converge to exact solution with optimal convergence rates under and norms on primal arbitrary triangular/quasi–parallelogram meshes. Finally, several numerical examples are carried out to validate the theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139836606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discrete null field equation methods for solving Laplace's equation: Boundary layer computations","authors":"Li-Ping Zhang, Zi‐Cai Li, Ming-Gong Lee, Hung‐Tsai Huang","doi":"10.1002/num.23092","DOIUrl":"https://doi.org/10.1002/num.23092","url":null,"abstract":"Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139777475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows","authors":"Yiwei Feng, Lili Lv, Tiegang Liu, Liang Xu, Weixiong Yuan","doi":"10.1002/num.23095","DOIUrl":"https://doi.org/10.1002/num.23095","url":null,"abstract":"Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139839445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows","authors":"Yiwei Feng, Lili Lv, Tiegang Liu, Liang Xu, Weixiong Yuan","doi":"10.1002/num.23095","DOIUrl":"https://doi.org/10.1002/num.23095","url":null,"abstract":"Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139779445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media","authors":"Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira","doi":"10.1002/num.23090","DOIUrl":"https://doi.org/10.1002/num.23090","url":null,"abstract":"This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139839694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media","authors":"Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira","doi":"10.1002/num.23090","DOIUrl":"https://doi.org/10.1002/num.23090","url":null,"abstract":"This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139779774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A second-order time discretizing block-centered finite difference method for compressible wormhole propagation","authors":"Fei Sun, Xiaoli Li, Hongxing Rui","doi":"10.1002/num.23091","DOIUrl":"https://doi.org/10.1002/num.23091","url":null,"abstract":"In this paper, a second-order time discretizing block-centered finite difference method is introduced to solve the compressible wormhole propagation. The optimal second-order error estimates for the porosity, pressure, velocity, concentration and its flux are established carefully in different discrete norms on non-uniform grids. Then by introducing Lagrange multiplier, a novel bound-preserving scheme for concentration is constructed. Finally, numerical experiments are carried out to demonstrate the correctness of theoretical analysis and capability for simulations of compressible wormhole propagation.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Retraction: Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps","authors":"","doi":"10.1002/num.23089","DOIUrl":"https://doi.org/10.1002/num.23089","url":null,"abstract":"<b>Retraction:</b> Nurgabyl DN, Uaissov AB. Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps. <i>Numer Methods Partial Differential Eq</i>. 2021; 37: 2375–2392. https://doi.org/10.1002/num.22719","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system","authors":"Yali Gao, Daozhi Han","doi":"10.1002/num.23087","DOIUrl":"https://doi.org/10.1002/num.23087","url":null,"abstract":"We develop two totally decoupled, linear and second-order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele-Shaw cell. The implicit-explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon).","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the convergence of 2D P4+ triangular and 3D P6+ tetrahedral divergence-free finite elements","authors":"Shangyou Zhang","doi":"10.1002/num.23088","DOIUrl":"https://doi.org/10.1002/num.23088","url":null,"abstract":"We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0003\" display=\"inline\" location=\"graphic/num23088-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {P}_k $$</annotation>\u0000</semantics></math>-<math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0004\" display=\"inline\" location=\"graphic/num23088-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msubsup>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000<mrow>\u0000<mtext>disc</mtext>\u0000</mrow>\u0000</msubsup>\u0000</mrow>\u0000$$ {P}_{k-1}^{mathrm{disc}} $$</annotation>\u0000</semantics></math> mixed finite element method for <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0005\" display=\"inline\" location=\"graphic/num23088-math-0005.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo>≥</mo>\u0000<mn>4</mn>\u0000</mrow>\u0000$$ kge 4 $$</annotation>\u0000</semantics></math> on 2D triangular grids or <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0006\" display=\"inline\" location=\"graphic/num23088-math-0006.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo>≥</mo>\u0000<mn>6</mn>\u0000</mrow>\u0000$$ kge 6 $$</annotation>\u0000</semantics></math> on tetrahedral grids, even in the case the inf-sup condition fails. By a simple <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0007\" display=\"inline\" location=\"graphic/num23088-math-0007.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mi>L</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000</msup>\u0000</mrow>\u0000$$ {L}^2 $$</annotation>\u0000</semantics></math>-projection of the discrete <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0008\" display=\"inline\" location=\"graphic/num23088-math-0008.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {P}_{k-1} $$</annotation>\u0000</semantics></math> pressure to the space of continuous <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0009\" display=\"inline\" location=\"graphic/num23088-math-0009.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {P}_{k-1} $$</annotation>\u0000</semantics></math> polynomials, we show this post-processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}