Бидний тухай
Багш ажилтан
Simulating incompressible lid-driven cavity flow requires solving the Navier–Stokes equations, where the pressure Poisson solve is often the dominant computational cost in pressure–velocity coupling schemes. This motivates Poisson solvers that are straightforward to implement, fast, and sufficiently accurate. In this work, we solve the two-dimensional Poisson equation on a square grid using a second-order five-point finite-difference discretization with homogeneous Neumann boundary conditions. We benchmark representative direct methods (dense-matrix LU factorization, sparse-matrix LU factorization, and a discrete cosine transform (DCT) approach) against simple stationary iterative methods (Jacobi, Gauss–Seidel, and successive overrelaxation). All solvers are implemented in Python 3.12 using NumPy, SciPy, and Numba in a Jupyter Notebook/Lab workflow on an Intel i7 quad-core laptop (4 CPUs, 8 threads, 2.3GHz base) with 16GB RAM under Windows. In our tests, the DCT-based solver is the fastest and most memory-efficient among the direct methods, followed by sparse LU. Among the iterative methods, Numba-accelerated SOR is the fastest when the relaxation parameter is chosen near its optimal value, but it remains slower and less accurate than the direct solvers considered here. The novelty of this work is a transparent, tool-specific benchmark of these standard methods on clearly documented commodity hardware, yielding concrete runtime and memory comparisons that can be reused in similar studies. Our focus is the practical selection of an easy-to-use Poisson solver that remains performant; integrating the selected solver into full incompressible cavity-flow simulations is left for future work.
Хөдөлдөг тагтай сав доторх үл шахагдагч шингэний хөдөлгөөнийг загварчлахын тулд Навье-Стоксын тэгшитгэлүүдийг бодох шаардлагатай. Эдгээр тэгшитгэлийг хурдны ба даралтын орны хувьд ойролцоолж шийдэх явцад, Пуассоны тэгшитгэлийг бодох ёстой болдог нь хамгийн их хугацаа зарцуулах алхам байдаг. Тиймээс, хэрэгжүүлэхэд нэн хялбар мөртлөө хангалттай хурдан ажиллаж, үр дүнг тохирох нарийвчлалтайгаар гаргаж өгөх арга ашиглах нь чухал. Бид Пуассоны тэгшитгэлийг, 2 хэмжээст дөрвөлжин тор дээр нэгэн төрлийн Нейманы захын нөхцөлтэйгээр төгсгөлөг ялгаврын аргуудаар ойролцоолон тооцоолж бодов. Дөрвөлжин торыг “жижиг” (17 × 17, 33 × 33, 65 × 65), “дунд” (129 × 129, 257 × 257), “том” (513 × 513, 1025 × 1025), “хэт том” (2049 × 2049) хэмжээтэй гэх мэт тохиолдлуудад авч үзлээ. Ингэхдээ, түгээмэл хэрэглэгдэгч шууд аргууд (шигүү болон сийрэг матрицын LU задаргаа, дискрет косинус хувиргалт (DCT)), хамгийн хялбар итератив (Якобийн, Гаусс-Зайделийн, хэт релаксацийн (over-relaxation)) аргуудыг харьцуулан судлав. Тооцооны програмуудыг Windows орчинд Python 3.12 хэл дээр Numpy, Scipy болон Numba модулийг ашиглан Jupyter Notebook/Lab системд бичлээ. Үр дүнгийн график дүрслэлийг Matplotlib модулийн тусламжтайгаар хийв. Тооцоо үйлдэхэд ашигласан процессор: Intel i7 Quad-Core, 4 CPUs, 8 Threads, 2.3 GHz (base), 16 Gb RAM. Тооцооны үр дүнгээс үзэхэд, жижиг торууд дээр шууд аргуудаас сийрэг матрицын LU задаргаа (sparse LU) амжилттай хурдан (< 10−2 с) ажиллаж байна. Сийрэг матрицыг хэрхэн “бүтээж” буй нь чухал. Санах ойд дарамт учруулахгүйгээр сийрэг матрицыг үүсгэж чадвал, дунд торууд дээр хангалттай хурдан (< 10−1 с), том торууд дээр боломжийн хурдан ажиллаж (< 1 с), тор хэт томроход санах ой хүрэлцэхгүй болж байна. Дунд, том, хэт том торуудын хувьд DCT арга хамгийн амжилттай (сийрэг матрицын аргаас дунджаар 2-3 дахин хурдан) бөгөөд, Лапласианы матрицтай ажиллахгүй тул санах ойд ч хэмнэлттэй байна. Итератив аргууд дотроос тухайлбал хэт релаксацийн аргыг Python Numba Just-In-Time (JIT) хэрэгслээр хөрвүүлсэн болон хөрвүүлээгүй хувилбаруудаар авч үзсэн болно. Энэ нь сийрэг матрицын аргыг ямар ч тохиолдолд хурд болон нарийвчлалаар гүйцэхгүй байгаа юм. Эдгээр үр дүн онолын гаргалгаануудтай тохирч байна. Энэхүү ажлын шинэлэг тал нь, бид нэгэн төрлийн Нейманы захын нөхцөл бүхий Пуассоны тэгшитгэлийг 2 хэмжээст дөрвөлжин тор дээр бодохын тулд, онолын хувьд сайтар судлагдсан хамгийн хялбар гэгдэх аргуудыг сонгон авч тодорхой техник үзүүлэлт бүхий персонал компьютер дээр тодорхой програмчлалын хэрэгсэл (Python + Numpy/Scipy/Numba-JIT) ашиглан туршилтаар (тооцоогоор) харьцуулан судалж, эдгээрийн дотроос хамгийн хурдан аргыг тодорхойлсон (DCT). Бидний гүйцэтгэсэн ажилд дурдагдсан бодит өгөгдлүүд (компьютерын үзүүлэлт, ашигласан програмчлалын технологи, тоон аргуудыг хэрхэн програмчилсан гэх мэт), гаргаж авсан тооцооны үр дүн (хугацаа) зэрэг нь цаашид иймэрхүү төрлийн бусад харьцуулсан судалгаанд баримжаа чиглүүлэг болох ач холбогдолтой гэж үзэж байна. Энэ удаа бид зөвхөн хэрэглэхэд хамгийн хялбар мөртлөө хангалттай хурд бүхий Пуассоны (тэгшитгэлийн) тооцоолуурыг сонгон судлах ажил дээр төвлөрсөн болно. Сонгож авсан тооцоолуураа хөдөлдөг тагтай сав доторх үл шахагдагч шингэний хөдөлгөөний загварчлалд хэрхэн ашиглах нь дараагийн шатны ажил байх юм.
This paper is concerned with the relaxation-time limits to a multidimensional radial steady hydrodynamic model of semiconductors in the form of Euler–Poisson equations with sonic or nonsonic boundary as the relaxation time and , respectively, where the sonic boundary is the critical and difficult case, because of the degeneracy at the boundary and the formation of boundary layers. For the case of , after showing the boundedness of the density by using the divergence form, we prove the convergence of the solutions to their nontrivial asymptotic states with the convergence order in the -sense. In order to overcome the degeneracy caused by the critical sonic boundary, we introduce an inverse transform as a technical tool to remove the second-order degeneracy, and observe the advantage of a first-order degeneracy due to the monotonicity of this transformation. Moreover, when with different boundary values, where the boundary layers appear, we show the strong convergence order or for different boundary cases. In order to overcome the difficulty caused by the boundary layer, we propose a new technique in asymptotic limit analysis and identify the width of the boundary layers as . These new proposed methods develop and improve upon the existing studies. Finally, a series of numerical simulations are conducted, which corroborate our theoretical analysis, particularly regarding the formation of boundary layers.
We establish a conditional optimality result for an adaptive mixed finite element method for the stationary Stokes problem discretized by the standard Taylor–Hood elements under the assumption of the so-called general quasiorthogonality. Optimality is measured in terms of a modified approximation class defined through the total error. We prove that the modified approximation class coincides with the standard approximation class, modulo the assumption that the data is regular enough in an appropriate scale of Besov spaces.
In this talk, we will present a new elementary approach to establishing elliptic estimates for a class of operators with rough coefficients, in the Triebel-Lizorkin and Besov scales. This is is a joint work with Mike Holst (UCSD) and David Maxwell (UAF).
In this talk, we establish a conditional optimality result for an adaptive mixed finite element method for the stationary Stokes problem discretized by the standard Taylor-Hood elements, under the assumption of the so-called general quasi-orthogonality. Optimality is measured in terms of a modified approximation class defined through the total error, as is customary since the seminal work of Cascon, Kreuzer, Nochetto and Siebert. Time permitting, the second part of the talk is independent of optimality results, and concerns interrelations between the modified approximation classes and the standard approximation classes (the latter defined through the energy error). Building on the tools developed in the papers of Binev, Dahmen, DeVore, and Petrushev, and of Gaspoz and Morin, we prove that the modified approximation class coincides with the standard approximation class, modulo the assumption that the data is regular enough in an appropriate scale of Besov spaces.
Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines 𝑟2𝑛 ln 𝑟 and 𝑟2𝑛+1 ‘‘failed to produce acceptable results’’. In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).
We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.
We will discuss a possible approach to establish elliptic estimates for opera- tors with barely continuous coefficients in a Sobolev-Besov and Triebel-Lizorkin scale. The result would obviously be not new but the proposed approach is relatively elemen- tary and therefore of interest. The whole theory depends on certain scaling properties of functions. We will discuss ways to establish those properties. This is a joint work with David Maxwell (UAF) and Michael Holst (UCSD).
Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge-Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking. We develop a general framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC for the scalar Poisson problem in arbitrary dimensions. This method is closely related to the lattice gauge theory discretization of the classical Yang-Mills equations, and therefore might shed some light on quantum gauge field theories in the future.
In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis func- tion (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method. The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the lit- erature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving non- linear elliptic PDEs.
In this talk we consider a general optimal control problem subject to PDE constraints, with the constraints defined in a general framework associated with Hilbert complexes. Hilbert complexes are a nested sequence of geometric spaces which is traversed by the application of differential operators and allows for generic analysis for operator equations. The framework is very general, but to give a standard example, the de Rham complex involves a finite sequence involving gradient, curl, and divergence operators. The motivation for considering this in the context of optimal control is twofold. First, typically, analysis in optimal control, and in particular PDE constrained optimization, involves, beyond a few very rudimentary results, a comprehensive often qualitatively unique line of arguments for each particular case. The Hilbert complex framework is endowed with sufficiently rich geometry such that informative results can be derived for a wide range of problems. Second, the rich geometric structure and its associated theoretical understanding is associated with finite element exterior calculus, again a general algorithmic analysis framework for discretization of the problem using finite dimensional subspaces with comprehensively powerful properties. This is an ongoing work joint with M.Holst (UCSD), V.Kungurtsev (CTU Prague), and M.Licht (EPFL).
