In this article, we construct first‐ and second‐order semidiscrete schemes for the magnetohydrodynamics (MHD) equations with variable density based on scalar auxiliary variable (SAV) approach. These schemes are decoupled, unconditionally energy stable and only solve a sequence of linear differential equations at each time step. We carry out a rigorous error analysis for the first‐order SAV scheme in two‐dimensional case. Some numerical experiments are presented to verify the accuracy and stability.
{"title":"Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density","authors":"Han Chen, Yuyu He, Hongtao Chen","doi":"10.1002/num.23067","DOIUrl":"https://doi.org/10.1002/num.23067","url":null,"abstract":"In this article, we construct first‐ and second‐order semidiscrete schemes for the magnetohydrodynamics (MHD) equations with variable density based on scalar auxiliary variable (SAV) approach. These schemes are decoupled, unconditionally energy stable and only solve a sequence of linear differential equations at each time step. We carry out a rigorous error analysis for the first‐order SAV scheme in two‐dimensional case. Some numerical experiments are presented to verify the accuracy and stability.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48996771","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}
Abstract In this article, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid‐structure interaction problems. Two finite element schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rates.
{"title":"Unfitted mixed finite element methods for elliptic interface problems","authors":"Najwa Alshehri, Daniele Boffi, Lucia Gastaldi","doi":"10.1002/num.23063","DOIUrl":"https://doi.org/10.1002/num.23063","url":null,"abstract":"Abstract In this article, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid‐structure interaction problems. Two finite element schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rates.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135397363","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}
The Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy‐preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the imaginary functions of Klein–Gordon–Schrödinger equations, and transform the original system into its real formulation. Based on the invariant energy quadratization approach, an equivalent system is deduced by introducing a Lagrange multiplier. Then the efficient and unconditionally stable scheme is designed to discretize the deduced equivalent system. A numerical analysis of the proposed scheme is presented to illustrate its uniquely solvability and convergence. Numerical examples are provided to validate accuracy, energy and mass conservation laws, and stability of our proposed method.
{"title":"An efficient linearly implicit and energy‐conservative scheme for two dimensional Klein–Gordon–Schrödinger equations","authors":"Hongwei Li, Yuna Yang, Xiangkun Li","doi":"10.1002/num.23064","DOIUrl":"https://doi.org/10.1002/num.23064","url":null,"abstract":"The Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy‐preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the imaginary functions of Klein–Gordon–Schrödinger equations, and transform the original system into its real formulation. Based on the invariant energy quadratization approach, an equivalent system is deduced by introducing a Lagrange multiplier. Then the efficient and unconditionally stable scheme is designed to discretize the deduced equivalent system. A numerical analysis of the proposed scheme is presented to illustrate its uniquely solvability and convergence. Numerical examples are provided to validate accuracy, energy and mass conservation laws, and stability of our proposed method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48827312","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}
In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings.
{"title":"Strong convergence for an explicit fully‐discrete finite element approximation of the Cahn‐Hillard‐Cook equation with additive noise","authors":"Qiu Lin, Ruisheng Qi","doi":"10.1002/num.23062","DOIUrl":"https://doi.org/10.1002/num.23062","url":null,"abstract":"In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44255083","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}
In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient‐type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank–Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration.
{"title":"Spatio‐temporal scalar auxiliary variable approach for the nonlinear convection–diffusion equation with discontinuous Galerkin method","authors":"Yaping Li, W. Zhao, Wenju Zhao","doi":"10.1002/num.23061","DOIUrl":"https://doi.org/10.1002/num.23061","url":null,"abstract":"In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient‐type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank–Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48751612","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}
In this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in H1$$ {H}^1 $$ ‐norm and pressure in L2$$ {L}^2 $$ ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.
{"title":"Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations","authors":"Huaijun Yang, Xu Jia","doi":"10.1002/num.23060","DOIUrl":"https://doi.org/10.1002/num.23060","url":null,"abstract":"In this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in H1$$ {H}^1 $$ ‐norm and pressure in L2$$ {L}^2 $$ ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43185276","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}
In this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two‐sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection‐diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two‐sided KPOD approach involving the block Arnoldi and block Lanczos processes for the obtained time‐varying equations is put forward. Moreover, another type of two‐sided KPOD algorithm based on Laguerre orthogonal polynomials in frequency domain is provided. For the two kinds of two‐sided KPOD methods, we present a theoretical analysis for the moment matching of the discrete time‐invariant transfer function in time domain and give the error bound caused by the reduced‐order projection between the Galerkin finite element solution and the approximate solution of the two‐sided KPOD method. Finally, the feasibility of four two‐sided KPOD algorithms is verified by several numerical results with different inputs and setting parameters.
{"title":"Two‐sided Krylov enhanced proper orthogonal decomposition methods for partial differential equations with variable coefficients","authors":"Li Wang, Zhen Miao, Yaolin Jiang","doi":"10.1002/num.23058","DOIUrl":"https://doi.org/10.1002/num.23058","url":null,"abstract":"In this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two‐sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection‐diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two‐sided KPOD approach involving the block Arnoldi and block Lanczos processes for the obtained time‐varying equations is put forward. Moreover, another type of two‐sided KPOD algorithm based on Laguerre orthogonal polynomials in frequency domain is provided. For the two kinds of two‐sided KPOD methods, we present a theoretical analysis for the moment matching of the discrete time‐invariant transfer function in time domain and give the error bound caused by the reduced‐order projection between the Galerkin finite element solution and the approximate solution of the two‐sided KPOD method. Finally, the feasibility of four two‐sided KPOD algorithms is verified by several numerical results with different inputs and setting parameters.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47673516","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}
This paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.
{"title":"A priori error estimates of two monolithic schemes for Biot's consolidation model","authors":"H. Gu, M. Cai, Jingzhi Li, Guoliang Ju","doi":"10.1002/num.23059","DOIUrl":"https://doi.org/10.1002/num.23059","url":null,"abstract":"This paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44862642","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}
The Neilan Pk$$ {P}_k $$ ‐ Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory.
{"title":"Neilan's divergence‐free finite elements for Stokes equations on tetrahedral grids","authors":"Shangyou Zhang","doi":"10.1002/num.23055","DOIUrl":"https://doi.org/10.1002/num.23055","url":null,"abstract":"The Neilan Pk$$ {P}_k $$ ‐ Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49393949","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}
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