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Computational Methods in Applied Mathematics最新文献

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Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity 具有对数非线性的半线性热方程的隐显有限差分逼近
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-31 DOI: 10.1515/cmam-2022-0217
Panagiotis Paraschis, G. E. Zouraris
{"title":"Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity","authors":"Panagiotis Paraschis, G. E. Zouraris","doi":"10.1515/cmam-2022-0217","DOIUrl":"https://doi.org/10.1515/cmam-2022-0217","url":null,"abstract":"Abstract We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) L^{infty}_{t}(L^{infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) L^{infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"695 - 713"},"PeriodicalIF":1.3,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49242850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Recent Advances in Boundary Element Methods 边界元方法的最新进展
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-28 DOI: 10.1515/cmam-2023-0037
U. Langer, O. Steinbach
{"title":"Recent Advances in Boundary Element Methods","authors":"U. Langer, O. Steinbach","doi":"10.1515/cmam-2023-0037","DOIUrl":"https://doi.org/10.1515/cmam-2023-0037","url":null,"abstract":"","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"297 - 299"},"PeriodicalIF":1.3,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41867304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Convenient Inclusion of Inhomogeneous Boundary Conditions in Minimal Residual Methods 最小残差法中一种方便的非齐次边界条件包含
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-22 DOI: 10.48550/arXiv.2303.12555
R. Stevenson
{"title":"A Convenient Inclusion of Inhomogeneous Boundary Conditions in Minimal Residual Methods","authors":"R. Stevenson","doi":"10.48550/arXiv.2303.12555","DOIUrl":"https://doi.org/10.48550/arXiv.2303.12555","url":null,"abstract":"Abstract Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain Ω on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on ∂ Ω partialOmega . In this paper, easily implementable minimal residual discretizations are constructed which yield quasi-optimal approximation from the employed trial space, in which the evaluation of fractional Sobolev norms is fully avoided.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"0 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41822558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements 有限元与边界元非对称耦合稳定性研究进展
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-10 DOI: 10.1515/cmam-2022-0085
M. Ferrari
{"title":"Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements","authors":"M. Ferrari","doi":"10.1515/cmam-2022-0085","DOIUrl":"https://doi.org/10.1515/cmam-2022-0085","url":null,"abstract":"Abstract We consider the non-symmetric coupling of finite and boundary elements to solve second-order nonlinear partial differential equations defined in unbounded domains. We present a novel condition that ensures that the associated semi-linear form induces a strongly monotone operator, keeping track of the dependence on the linear combination of the interior domain equation with the boundary integral one. We show that an optimal ellipticity condition, relating the nonlinear operator to the contraction constant of the shifted double-layer integral operator, is guaranteed by choosing a particular linear combination. These results generalize those obtained by Of and Steinbach [Is the one-equation coupling of finite and boundary element methods always stable?, ZAMM Z. Angew. Math. Mech. 93 (2013), 6–7, 476–484] and [On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems, Numer. Math. 127 (2014), 3, 567–593], and by Steinbach [A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal. 49 (2011), 4, 1521–1531], where the simple sum of the two coupling equations has been considered. Numerical examples confirm the theoretical results on the sharpness of the presented estimates.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"373 - 388"},"PeriodicalIF":1.3,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41847506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems 奇异非线性传动与接触问题的有限元与边界元耦合
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-09 DOI: 10.1515/cmam-2022-0120
H. Gimperlein, E. Stephan
{"title":"Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems","authors":"H. Gimperlein, E. Stephan","doi":"10.1515/cmam-2022-0120","DOIUrl":"https://doi.org/10.1515/cmam-2022-0120","url":null,"abstract":"Abstract This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for interface problems in nonlinear elasticity. It concerns 𝑝-Laplacian-type Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"389 - 404"},"PeriodicalIF":1.3,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41328644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries 复杂几何障碍物散射时域波场的cem - bem耦合模拟
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-09 DOI: 10.1515/cmam-2022-0084
L. Desiderio, S. Falletta, M. Ferrari, L. Scuderi
{"title":"CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries","authors":"L. Desiderio, S. Falletta, M. Ferrari, L. Scuderi","doi":"10.1515/cmam-2022-0084","DOIUrl":"https://doi.org/10.1515/cmam-2022-0084","url":null,"abstract":"Abstract In this paper, we present a numerical method based on the coupling between a Curved Virtual Element Method (CVEM) and a Boundary Element Method (BEM) for the simulation of wave fields scattered by obstacles immersed in homogeneous infinite media. In particular, we consider the 2D time-domain damped wave equation, endowed with a Dirichlet condition on the boundary (sound-soft scattering). To reduce the infinite domain to a finite computational one, we introduce an artificial boundary on which we impose a Boundary Integral Non-Reflecting Boundary Condition (BI-NRBC). We apply a CVEM combined with the Crank–Nicolson time integrator in the interior domain, and we discretize the BI-NRBC by a convolution quadrature formula in time and a collocation method in space. We present some numerical results to test the performance of the proposed approach and to highlight its effectiveness, especially when obstacles with complex geometries are considered.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"353 - 372"},"PeriodicalIF":1.3,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45931680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Force Computation for Dielectrics Using Shape Calculus 用形状微积分计算电介质的力
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-08 DOI: 10.1515/cmam-2022-0112
P. Panchal, N. Ren, R. Hiptmair
{"title":"Force Computation for Dielectrics Using Shape Calculus","authors":"P. Panchal, N. Ren, R. Hiptmair","doi":"10.1515/cmam-2022-0112","DOIUrl":"https://doi.org/10.1515/cmam-2022-0112","url":null,"abstract":"Abstract We are concerned with the numerical computation of electrostatic forces/torques in only piece-wise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered in [P. Panchal and R. Hiptmair, Electrostatic force computation with boundary element methods, SMAI J. Comput. Math. 8 (2022), 49–74]. Motivated by the virtual work principle which is interpreted using techniques of shape calculus, and using the adjoint method from shape optimization, we derive stable interface-based force functionals suitable for use with BEM. This is done in the framework of single-trace direct boundary integral equations for second-order transmission problems. Numerical tests confirm the fast asymptotic convergence and superior accuracy of the new formulas for the computation of total forces and torques.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"425 - 444"},"PeriodicalIF":1.3,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48145609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints 带不动点约束的多输出集的分裂单调变分包含问题
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-03-01 DOI: 10.1515/cmam-2022-0199
V. A. Uzor, T. O. Alakoya, O. Mewomo
{"title":"On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints","authors":"V. A. Uzor, T. O. Alakoya, O. Mewomo","doi":"10.1515/cmam-2022-0199","DOIUrl":"https://doi.org/10.1515/cmam-2022-0199","url":null,"abstract":"Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"729 - 749"},"PeriodicalIF":1.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42365646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method 基于拟合L1方法的延迟分数阶方程导数不连续跟踪
IF 1.3 4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-02-28 DOI: 10.1515/cmam-2022-0231
Dakang Cen, Seakweng Vong
{"title":"The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method","authors":"Dakang Cen, Seakweng Vong","doi":"10.1515/cmam-2022-0231","DOIUrl":"https://doi.org/10.1515/cmam-2022-0231","url":null,"abstract":"Abstract In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + {s^{+}} is better than that at 0 + {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L ⁢ 1 {L1} numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [ 2 3 , 1 ) {alphain[frac{2}{3},1)} , α is the order of fractional derivative. Furthermore, an improved fitted L ⁢ 1 {L1} method is proposed and the region of optimal convergence order is larger. For the case t > s {t>s} , stability and min ⁡ { 2 ⁢ α , 1 } {min{2alpha,1}} order convergence of the fitted L ⁢ 1 {L1} scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":"591 - 601"},"PeriodicalIF":1.3,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45965201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems 分布椭圆型最优控制问题的鲁棒有限元离散化及求解方法
4区 数学
Computational Methods in Applied Mathematics Pub Date : 2023-02-18 DOI: 10.1515/cmam-2022-0138
Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang
{"title":"Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems","authors":"Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang","doi":"10.1515/cmam-2022-0138","DOIUrl":"https://doi.org/10.1515/cmam-2022-0138","url":null,"abstract":"Abstract We consider standard tracking-type, distributed elliptic optimal control problems with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} regularization, and their finite element discretization. We are investigating the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} error between the finite element approximation <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>ϱ</m:mi> <m:mo>⁢</m:mo> <m:mi>h</m:mi> </m:mrow> </m:msub> </m:math> u_{varrho h} of the state <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>u</m:mi> <m:mi>ϱ</m:mi> </m:msub> </m:math> u_{varrho} and the desired state (target) <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mi>u</m:mi> <m:mo>¯</m:mo> </m:mover> </m:math> overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϱ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135285323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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