IMA Journal of Numerical Analysis最新文献

{"title":"Error analysis for local discontinuous Galerkin semidiscretization of Richards’ equation","authors":"Scott Congreve, Vít Dolejší, Sunčica Sakić","doi":"10.1093/imanum/drae013","DOIUrl":"https://doi.org/10.1093/imanum/drae013","url":null,"abstract":"This paper concerns an error analysis of the space semidiscrete scheme for the Richards’ equation modeling flows in variably saturated porous media. This nonlinear parabolic partial differential equation can degenerate; namely, we consider the case where the time derivative term can vanish, i.e., the fast-diffusion type of degeneracy. We discretize the Richards’ equation by the local discontinuous Galerkin method, which provides high order accuracy and preserves stability. Due to the nonlinearity of the problem, special techniques for numerical analysis of the scheme are required. In particular, we combine two partial error bounds using continuous mathematical induction and derive a priori error estimates with respect to the spatial discretization parameter and the Hölder coefficient of the nonlinear temporal derivative. Finally, the theoretical results are supported by numerical experiments, including cases beyond the assumptions of the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"44 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140910600","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 linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients","authors":"Mengchao Wang, Xiaojie Wang","doi":"10.1093/imanum/drae012","DOIUrl":"https://doi.org/10.1093/imanum/drae012","url":null,"abstract":"The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ mathcal{D} in{mathbb{R}}^{d}$, $ d leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($text{Tr} (Q) < infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"50 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895675","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":"Stability of convergence rates: kernel interpolation on non-Lipschitz domains","authors":"Tizian Wenzel, Gabriele Santin, Bernard Haasdonk","doi":"10.1093/imanum/drae014","DOIUrl":"https://doi.org/10.1093/imanum/drae014","url":null,"abstract":"Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $varOmega subset{mathbb{R}} ^{d}$, thus allowing for non-Lipschitz domains including e.g., cusps and irregular boundaries. Especially we show that, when going to a smaller domain $tilde{varOmega } subset varOmega subset{mathbb{R}} ^{d}$, the convergence rate does not deteriorate—i.e., the convergence rates are stable with respect to going to a subset. We obtain this by leveraging an analysis of greedy kernel algorithms. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $varOmega $ has an impact on the approximation properties. Numerical experiments illustrate and confirm the analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895686","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":"Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation","authors":"Chuchu Chen, Tonghe Dang, Jialin Hong","doi":"10.1093/imanum/drae009","DOIUrl":"https://doi.org/10.1093/imanum/drae009","url":null,"abstract":"It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $mathscr{C}(mathscr{O})$-norm and $dot{H}^{beta }(mathscr{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $frac{beta }{2}$ in time and $frac{beta }{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $beta in (0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826360","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 certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations","authors":"Lewin Ernst, Karsten Urban","doi":"10.1093/imanum/drae011","DOIUrl":"https://doi.org/10.1093/imanum/drae011","url":null,"abstract":"Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"125 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826237","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":"Finite element methods for multicomponent convection-diffusion","authors":"Francis R A Aznaran, Patrick E Farrell, Charles W Monroe, Alexander J Van-Brunt","doi":"10.1093/imanum/drae001","DOIUrl":"https://doi.org/10.1093/imanum/drae001","url":null,"abstract":"We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140819116","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":"An explicit spectral Fletcher–Reeves conjugate gradient method for bi-criteria optimization","authors":"Y Elboulqe, M El Maghri","doi":"10.1093/imanum/drae003","DOIUrl":"https://doi.org/10.1093/imanum/drae003","url":null,"abstract":"In this paper, we propose a spectral Fletcher–Reeves conjugate gradient-like method for solving unconstrained bi-criteria minimization problems without using any technique of scalarization. We suggest an explicit formulae for computing a descent direction common to both criteria. The latter further verifies a sufficient descent property that does not depend on the line search nor on any convexity assumption. After proving the existence of a bi-criteria Armijo-type stepsize, global convergence of the proposed algorithm is established. Finally, some numerical results and comparisons with other methods are reported.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"169 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140550430","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":"On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation","authors":"Piotr Gwiazda, Jakub Skrzeczkowski, Lara Trussardi","doi":"10.1093/imanum/drae006","DOIUrl":"https://doi.org/10.1093/imanum/drae006","url":null,"abstract":"It is well-known that one can construct solutions to the nonlocal Cahn–Hilliard equation with singular potentials via Yosida approximation with parameter $lambda to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $sqrt{lambda }$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert–Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $lambda $ could be linked to the discretization parameters, yielding appropriate error estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"79 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140544700","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":"Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media","authors":"Yanzhao Cao, Thi-Thao-Phuong Hoang, Phuoc-Toan Huynh","doi":"10.1093/imanum/drae005","DOIUrl":"https://doi.org/10.1093/imanum/drae005","url":null,"abstract":"The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140349099","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":"On the necessity of the inf-sup condition for a mixed finite element formulation","authors":"Fleurianne Bertrand, Daniele Boffi","doi":"10.1093/imanum/drae002","DOIUrl":"https://doi.org/10.1093/imanum/drae002","url":null,"abstract":"We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $textbf{H}(operatorname{textrm{div}};varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140000943","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}