Computers & Mathematics with Applications最新文献

{"title":"Analysis of new mixed finite element method for a Barenblatt-Biot poroelastic model","authors":"","doi":"10.1016/j.camwa.2024.07.011","DOIUrl":"10.1016/j.camwa.2024.07.011","url":null,"abstract":"<div><p>In this work, we study the locking-free numerical method for a Barenblatt-Biot poroelastic model. When solving by the continuous Galerkin mixed finite element method, the model exists two kind of locking phenomena for special physical parameters. To overcome these locking phenomena, we introduce new variables to reformulate the original problem into a new problem, which exists a built-in mechanism to keep the continuous Galerkin mixed finite element method stable. It can be regarded as a generalized Stokes problem for two given <em>β</em> and <em>η</em> and two diffusion problems for a given <em>δ</em>. The generalized Stokes problem can be adopted by some stable solver and the diffusion problems can be solved by continuous Galerkin finite element. Moreover, the existence and uniqueness of weak solution is proved by using the standard Galerkin method and combining with priori estimates and some invariant quantities. After that, we design fully discrete time-stepping schemes to use mixed finite element method with <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> element pairs for space variables and backward Euler method for time variable, and analyze the coupled and decoupled time stepping methods based on the proposed scheme. The optimal convergence order is obtained in both space and time. Finally, some numerical examples are presented to show the optimal convergence rates about variables and the robustness of proposed method with respect to <em>ν</em>, and to verify that there is no locking phenomenon.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New quadratic and cubic polynomial enrichments of the Crouzeix–Raviart finite element","authors":"","doi":"10.1016/j.camwa.2024.06.019","DOIUrl":"10.1016/j.camwa.2024.06.019","url":null,"abstract":"<div><p>In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix–Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessary and sufficient condition under which these augmented elements are well-defined. For illustration purposes, we then use a general approach to define two-parameter families of admissible degrees of freedom. Additionally, we provide explicit expressions for the associated basis functions and subsequently introduce new quadratic and cubic approximation operators based on the proposed admissible elements. The efficiency of the enriched methods is compared with that of the triangular Crouzeix–Raviart element. As expected, the numerical results exhibit a significant improvement, confirming the effectiveness of the developed enrichment strategy.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0898122124002876/pdfft?md5=efb59dbb046dc854aeaade45963dc4e0&pid=1-s2.0-S0898122124002876-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141639307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical analysis of time filter method for the stabilized incompressible diffusive Peterlin viscoelastic fluid model","authors":"","doi":"10.1016/j.camwa.2024.07.002","DOIUrl":"10.1016/j.camwa.2024.07.002","url":null,"abstract":"<div><p>The Diffusion Peterlin Viscoelastic Fluid (DPVF) model describes the movement of specific incompressible polymeric fluids. In this paper, we introduce and evaluate a new low-complexity linear-time filter finite element (FE) method for the DPVF model. In order to avoid the value at time <span><math><mi>t</mi><mo>=</mo><mo>−</mo><mi>Δ</mi><mi>t</mi></math></span>, the proposed time filter method consists of three steps, including a post-processing step. Firstly, a first-order Euler backward nonlinear fully discrete mixed FE scheme is employed to compute the numerical solutions at time <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>Δ</mi><mi>t</mi></math></span>. For <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, we obtain the intermediate values <span><math><mo>(</mo><msubsup><mrow><mover><mrow><mtext>u</mtext></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mover><mrow><mtext>d</mtext></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></math></span> in Step II using a fully implicit backward Euler scheme. At the same time level, we proceed with these intermediate values <span><math><mo>(</mo><msubsup><mrow><mover><mrow><mtext>u</mtext></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mover><mrow><mtext>d</mtext></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></math></span> using the linear time filters. The linear time filters step does not significantly increase computational complexity. However, it can enhance temporal convergence accuracy from first order to second order for backward Euler time filter (BE time filter), and from second order to three order for BDF2 time filter. We demonstrate the almost unconditional stability of the scheme. Error estimates for the time filter method are derived and presented. Several numerical experiments are conducted to validate the theoretical findings and showcase the efficiency of the proposed method.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141638036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A comparative numerical study of finite element methods resulting in mass conservation for Poisson's problem: Primal hybrid, mixed and their hybridized formulations","authors":"","doi":"10.1016/j.camwa.2024.06.027","DOIUrl":"10.1016/j.camwa.2024.06.027","url":null,"abstract":"<div><p>This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT_<em>k</em> space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141639308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Heat transfer effect on the ferrofluid flow in a curved cylindrical annular duct under the influence of a magnetic field","authors":"","doi":"10.1016/j.camwa.2024.06.026","DOIUrl":"10.1016/j.camwa.2024.06.026","url":null,"abstract":"<div><p>The current research, which can be employed in various engineering applications, is involved with the investigation of the heat transfer effect on the laminar and fully developed ferrohydrodynamic flow into a curved annular cylindrical duct, when a constant very strong transverse magnetic field is applied. The numerical solution of the involved constitutive partial differential equations, i.e. the continuity, momentum, energy, magnetization and Maxwell's equations with the corresponding boundary conditions, is achieved via the computational Continuity-Vorticity-Pressure (C.V.P.) algorithmic method, using a conveniently chosen non-uniform grid. The method is implemented via an in-house code, which has been applied and validated in several ferrohydrodynamic flows. It incorporates a general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids. The results show that the velocity distribution, the pressure drop and the temperature are significantly affected by the magnetic field strength and the volumetric concentration of the ferrofluid particles. The flow in the axial direction is redistributed in four symmetric poles, where its maximum value is readily observed. A secondary flow is generated, due to the combined effect of the buoyancy, the curvature and the magnetic field, which improves the heat transfer between the walls and the fluid. The axial pressure gradient, which is required to maintain the same mass flow, also increases as the field strength and concentration of magnetic particles increases, but with a lower rate in comparison to the increase in heat transfer.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141639309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A finite element contour integral method for computing the resonances of metallic grating structures with subwavelength holes","authors":"","doi":"10.1016/j.camwa.2024.06.022","DOIUrl":"10.1016/j.camwa.2024.06.022","url":null,"abstract":"<div><p>We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical model is multiscale due to the existence of several length scales in problem geometry and material contrast. We discretize the partial differential equation model over the truncated domain using the finite element method and develop a multi-step contour integral eigensolver to compute the resonances. The eigensolver first locates eigenvalues using a spectral indicator and then computes eigenvalues by a subspace projection scheme. The proposed numerical method is robust and scalable, and does not require initial guess as the iteration methods. Numerical examples are presented to demonstrate its effectiveness.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141639305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of variable-time-step BDF2 combined with the fast two-grid finite element algorithm for the FitzHugh-Nagumo model","authors":"","doi":"10.1016/j.camwa.2024.07.001","DOIUrl":"10.1016/j.camwa.2024.07.001","url":null,"abstract":"<div><p>In this article, a fast numerical method is developed for solving the FitzHugh-Nagumo (FHN) model by combining two-grid finite element (TGFE) algorithm in space with a linearized variable-time-step (VTS) two-step backward differentiation formula (BDF2) in time. This algorithm mainly included two steps: firstly, the nonlinear coupled system on the coarse grid is solved by a nonlinear iteration; secondly, a linearized coupled system on the fine grid by making use of a Taylor formula is formulated, and the numerical solution pair is solved directly. The techniques of the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels are used to derive the optimal error estimations in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and the stability analysis for the fully discrete scheme on the coarse and fine grids, and to prove the optimal <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm error estimation for the fully discrete TGFE scheme. These analyses hold for adjacent time-step ratios <span><math><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mn>4.8645</mn><mo>−</mo><mi>δ</mi></math></span> (<em>δ</em> being an arbitrarily small constant). Finally, the effectiveness and computing efficiency of the proposed algorithm are verified through several numerical examples.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141639306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Adaptive sampling points based multi-scale residual network for solving partial differential equations","authors":"","doi":"10.1016/j.camwa.2024.06.029","DOIUrl":"10.1016/j.camwa.2024.06.029","url":null,"abstract":"<div><p>Physics-informed neural networks (PINNs) have shown remarkable achievements in solving partial differential equations (PDEs). However, their performance is limited when encountering oscillatory part in the solutions of PDEs. Therefore, this paper proposes a multi-scale deep neural network with periodic activation function to achieve high-frequency to low-frequency conversion, which can capture the oscillation part of the solution of PDEs. Moreover, the use of adaptive sampling method can adaptively change the location and distribution of residual points, improving the performance of the network. Additionally, the gradient-enhanced strategy is also utilized to embed the gradient information of the PDEs into the loss function of the neural network, which further improves the accuracy of PINNs. Through the numerical experiments verification, it is found that our method is better than PINNs in terms of accuracy and efficiency.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence of adaptive mixed interior penalty discontinuous Galerkin methods for H(curl)-elliptic problems","authors":"","doi":"10.1016/j.camwa.2024.06.020","DOIUrl":"10.1016/j.camwa.2024.06.020","url":null,"abstract":"<div><p>In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for <span><math><mi>H</mi><mo>(</mo><mrow><mi>c</mi><mi>u</mi><mi>r</mi><mi>l</mi></mrow><mo>)</mo></math></span>-elliptic problems. We first get the mixed model of <span><math><mi>H</mi><mo>(</mo><mrow><mi>c</mi><mi>u</mi><mi>r</mi><mi>l</mi></mrow><mo>)</mo></math></span>-elliptic problem by introducing a new intermediate variable. Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation. Next, we construct the corresponding posteriori error indicator, and prove the contraction of the summation of the energy error and the scaled error indicator. At last, we confirm and illustrate the theoretical result through some numerical experiments.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A high-order arbitrary Lagrangian-Eulerian discontinuous Galerkin method for compressible flows in two-dimensional Cartesian and cylindrical coordinates","authors":"","doi":"10.1016/j.camwa.2024.06.021","DOIUrl":"10.1016/j.camwa.2024.06.021","url":null,"abstract":"<div><p>In this paper, a high-order direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is developed for compressible fluid flows in two-dimensional (2D) Cartesian and cylindrical coordinates. The scheme in 2D cylindrical coordinates is based on the control volume approach and it can preserve the conservation property for all the conserved variables including mass, momentum and total energy. In this hydrodynamic scheme, a kind of high-order Taylor expansion basis function on the general element is used to construct the interpolation polynomials of the physical variables for the DG discretization. The terms including the material derivatives of the test functions are omitted, which simplifies the scheme significantly. Furthermore, the mesh velocity in the direct ALE framework is obtained by implementing an adaptive mesh movement method with a kind of dimensional-splitting type monitor function. This type of mesh movement method can automatically concentrate the mesh nodes near the regions with large gradients of the variables, which can greatly improve the resolutions of numerical solutions near the specified regions. For removing the numerical oscillations in the simulations, a Hermite Weighted Essential Non-oscillatory (HWENO) reconstruction is employed as a slope limiter. Finally, some test cases are displayed to verify the accuracy and the good performance of our scheme.</p></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}