Search

In this paper, we study an elliptic variational problem regarding the p$$ p $$‐fractional Laplacian in ℝN$$ {\mathrm{\mathbb{R}}}^N $$ on the basis of recent result which generalizes some nice published work, and then give some sufficient conditions under which some weak solutions to our studied elliptic variational problem are continuous in ℝN$$ {\mathrm{\mathbb{R}}}^N $$. In the final appendix, we correct the proofs of two published lemmas for 1

This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the hybrid high‐order method. Combinations of h$$ h $$‐, p$$ p $$‐, and hp$$ hp $$‐coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non‐nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution. [ABSTRACT FROM AUTHOR]

By a probabilistic method, we prove the existence and uniqueness of weak solutions to Neumann problems for a class of semi-linear elliptic partial differential equations with nonlinear singular divergence terms, which can only be understood in distributional sense. This leads to the further study on a new class of infinite horizon backward stochastic differential equations, which involves integrals with respect to a forward–backward martingale and a singular continuous increasing process. [ABSTRACT FROM AUTHOR]

In this note we discuss results regarding the convergence in the sense of Mosco of a sequence of open sets. This concept of convergence of sets is a tool in the study of the convergence in Sobolev spaces of the solutions of an elliptic boundary value problem, as the domain is varying. [ABSTRACT FROM AUTHOR]

A high order modified nodal bi‐cubic spline collocation method is proposed for numerical solution of second‐order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coefficients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coefficients as long as all the requirements of the laws of physics are satisfied. To check the validity and accuracy of the proposed numerical method, some specific test problems with known solutions are solved. [ABSTRACT FROM AUTHOR]

We present a RBF-based semi-analytical technique for solving 3D convection–diffusion–reaction (CDR) equations to model transport in an anisotropic inhomogeneous medium. The mathematical model is expressed as the boundary value problem for elliptic partial differential equation (EPDE). Main feature of the presented technique is the separately satisfaction of the conditions on the boundary of the domain and the EPDE inside. To be more precise, we transform the original EPDE to the equation with homogeneous boundary condition (BC) and seek the approximate solution as a sum of the modified RBFs (MRBFs). The MRBFs satisfy the homogeneous BC of the problem. So, any linear combination also satisfies the homogeneous BC. The RBFs of three types are used in the framework of the method: the Multiquadric (MQ) RBF, the Gaussian RBF and the conical one. The coefficients of the linear combination are determined so that it satisfies the governing equation of the EPDE. Ten numerical examples demonstrate the high effectiveness of the presented technique in solving 3D CDR problems in single and double connected domains. [ABSTRACT FROM AUTHOR]

We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradient of an arbitrary solution at given height to that of a symmetric solution on a warped product model space. We also discuss the problem on Finsler manifolds with nonnegative weighted Ricci curvature, and on complete manifolds with bounded geometry, including solutions on manifolds with boundary with Dirichlet boundary condition. Particular cases of our results include gradient estimates of Modica type. [ABSTRACT FROM AUTHOR]

In this paper, two classes of methods are developed for the solution of two-dimensional elliptic partial differential equations. We have used tension spline function approximation in both x and y spatial directions and a new scheme of order O (h 4 + k 4) has been obtained. The convergence analysis of the methods has been carried out. Numerical examples are given to illustrate the applicability and accurate nature of our approach. [ABSTRACT FROM AUTHOR]

At present, the quality of online video courses in China is mixed. There are several reasons for the quality of online video courses. 1. The advantages and disadvantages of the front-end video capture equipment itself; 2. The distance of online video transmission; 3. The medium through which the video is transmitted; 4. Watch whether there is relevant interference information in the signal where the video is located and whether the video is compressed during transmission. These reasons lead to that although there is much to learn in the video, the resolution is too low to see from the video. With the development of the current social environment, most of the courses need online teaching. Therefore, in order to improve some problems in video playing caused by the increase of online teaching amount caused by the current environment, this paper provides higher resolution video for online courses by using high-resolution image processing technology based on the elliptic partial differential equation online video course. The high resolution processing technology used in this paper is centered on filtering algorithm. On the basis of the existing online video course of elliptic partial differential equations, the use of high-resolution technology can overcome the resolution limit of the hardware itself and further improve the video quality of online video teaching. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a c-stable linear equation in the tensor product space [ABSTRACT FROM AUTHOR]

We deal with jointly convex generalized Nash equilibrium problems in infinite-dimensional spaces. For their solution, we extend a finite-dimensional optimization approach and design a convergent algorithm in Hilbert space. Then we apply our investigations to a class of multiobjective optimal control problems with control and state constraints that are governed by elliptic partial differential equations. We present a new reformulation as a jointly convex generalized Nash equilibrium problem. We study a finite element approximation of such a multiobjective optimal control problem, and further we prove convergence in appropriate function spaces. Finally, we provide some numerical results that show the effectiveness of our algorithm for multiobjective optimal control problems. [ABSTRACT FROM AUTHOR]

This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution u and the derivatives u x , u x x and the mixed derivative u x y. The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme. [ABSTRACT FROM AUTHOR]

We discuss numerical reduction methods for an optimal control problem of semi-infinite type with finitely many control parameters but infinitely many constraints. We invoke known a priori error estimates to reduce the number of constraints. In a first strategy, we apply uniformly refined meshes, whereas in a second more heuristic strategy we use adaptive mesh refinement and provide an a posteriori error estimate for the control based on perturbation arguments. [ABSTRACT FROM AUTHOR]

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low. [ABSTRACT FROM AUTHOR]

We consider inverse problems of source identification in electroencephalography, modelled by elliptic partial differential equations. Being given boundary data that consist in values of the current flux and of the electric potential on the scalp, the aim is to reconstruct unknown current sources supported within the brain. For spherical layered models of the head, and after a preliminary data transmission step, such inverse source problems are tackled using best rational approximation techniques on planar sections. Both theoretical and constructive aspects are described, while numerical illustrations are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff-Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take into account the singularities. [ABSTRACT FROM AUTHOR]

We prove quantitative unique continuation results for solutions of −Δu + W · ∇u + Vu = λu, where λ ∈ ℂ andVandWare complex-valued decaying potentials that satisfy V(x) < ⟨x⟩−Nand W(x) < ⟨x⟩−P. ForM(R) = infx 0 = R‖u‖L 2(B 1(x 0)), we show that if the solutionuis non-zero, bounded, andu(0) = 1, thenM(R) ≳ exp(−CRβ0(logR)A(R)), where. Under certain conditions onN,Pand λ, we construct examples (some of which are in the style of Meshkov) to prove that this estimate forM(R) is sharp. That is, we construct functionsu,VandWsuch that −Δu + W · ∇u + Vu = λu, V(x) < ⟨x⟩−N, W(x) < ⟨x⟩−Pand u(x) xβ0(log x)C). [ABSTRACT FROM AUTHOR]

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions. [ABSTRACT FROM AUTHOR]

We consider shape optimization problems, where the state is governed by elliptic partial differential equations. Using a regularization technique, unknown shapes are encoded via shape functions, turning the shape optimization into optimal control problems for the unknown functions. The method is studied for elliptic PDEs to be solved in an unknown region (to be optimized), where the regularization technique together with a penalty method extends the PDE to a larger fixed domain. Additionally, the method is studied for the optimal layout problem, where the unknown regions determine the coefficients of the state equation. In both cases, the existence of optimal shapes is established for the regularized and for the original problem, with convergence of optimal shapes if the regularization parameter tends to zero. Error estimates are proved for the layout problem. In the context of finite element approximations, convergence and differentiability properties are shown. The method is designed to allow topological changes in a natural way, which is illustrated in a series of numerical experiments, applying the method to an elliptic PDE arising from an oil industry application with two unknown shapes, one giving the region where the PDE is solved, and the other determining the PDE's coefficients. [ABSTRACT FROM AUTHOR]

Abstract: Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems. [Copyright &y& Elsevier]

Optimal control problems in measure spaces governed by elliptic equations are considered for distributed and Neumann boundary control, which are known to promote sparse solutions. Optimality conditions are derived and some of the structural properties of their solutions, in particular sparsity, are discussed. A framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates. [ABSTRACT FROM AUTHOR]

We discuss the occurrence of positive solutions which decay to 0 as |x| → +∞ to the differential equation Δu + f(x, u) + g(|x|)x · ∇u = 0, |x| > R > 0, x ∈ ℝ n , where n ≥ 3, g is nonnegative valued and f has alternating sign, by means of the comparison method. Our results complement several recent contributions from Ehrnström and Mustafa [M. Ehrnström, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), pp. 1147–1154]. [ABSTRACT FROM AUTHOR]

We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ϵ-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ϵ-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270]. [ABSTRACT FROM AUTHOR]

Abstract: By incorporating the Legendre multiwavelet into the discontinuous Galerkin (DG) method, this paper presents a novel approach for solving Poisson’s equation with Dirichlet boundary, which is known as the discontinuous Legendre multiwavelet element (DLWE) method, derive an adaptive algorithm for the method, and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. Furthermore, this paper generalizes the DLWE method to the general elliptic equations defined on a bounded domain and describes the possibilities of constructing optimal adaptive algorithm. The proposed method and its generalizations are also applicable to some other kinds of partial differential equations. [Copyright &y& Elsevier]

A method for solving systems of linear algebraic equations arising in connection with the approximation of boundary value problems for elliptic partial differential equations is proposed. This method belongs to the class of conjugate directions method applied to a preliminary transformed system of equations. A model example is used to explain the idea underlying this method and to investigate it. Results of numerical experiments that confirm the method's efficiency are discussed. [ABSTRACT FROM AUTHOR]

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs. [ABSTRACT FROM AUTHOR]

Abstract: A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA–GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of -error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results. [Copyright &y& Elsevier]

Abstract: We propose a simple and effective numerical procedure for solving elliptic problems with non-Gaussian random coefficients, assuming that samples of the non-Gaussian random inputs are available from a statistical model. Given a correlation function, the Karhunen–Loève (K–L) expansion is employed to reduce the dimensionality of random inputs. Using the kernel density estimation technique, we obtain the marginal probability density functions (PDFs) of the random variables in the K–L expansion, based on which we define an auxiliary joint PDF. We then implement the generalized polynomial chaos (gPC) method via a collocation projection according to the auxiliary joint PDF. Based on the observation that the solution has an analytic extension in the parametric space, we ensure that the polynomial interpolation achieves point-wise convergence in the parametric space regardless of the PDF, where the energy norm is employed in the physical space. Hence, we can sample the gPC solution using the joint PDF instead of the auxiliary one to obtain the correct statistics. We also implement Monte Carlo methods to further refine the statistics using the gPC solution for variance reduction. Numerical results are presented to demonstrate the efficiency of the proposed approach. [Copyright &y& Elsevier]

Abstract: We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates. [Copyright &y& Elsevier]

A class of sufficient conditions are obtained for the existence and uniqueness of solutions to the boundary value problems of semi-linear elliptic partial differential equations, using a global inverse function theorem. [ABSTRACT FROM AUTHOR]

Abstract: The well-known Jaeger–Rosenthal asymptotic particular solution for the quasi-steady-state problem of moving heat source is proven to be inconsistent with the source constant intensity, especially at dimensionless trailing edge coordinates vx/a <−2. The problem is reduced to an equivalent Poisson’s equation by exponential transformation of moving coordinate scale. Using the method of images, the fundamental solution is found; the temperature rise function exponentially approximates to 0 along negative semi-axis. The temperature field in a semi-infinite solid for the general case of surface power intensity distribution is expressed, using the found Green’s function. The cases of point, line, and circular heat sources are considered. The found fundamental solution and particular solution for moving circular heat source explain the phenomena of martensite transformation in low-carbon steel substrate at relatively low source velocity 1.7cm/s. [Copyright &y& Elsevier]

We prove uniqueness in the Faber-Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that among all sufficiently smooth domains of fixed volume, the ball is the unique minimizer for the first eigenvalue. The method of proof, which avoids the use of any symmetrization, also works in the case of Dirichlet boundary conditions. We also give a characterization of all symmetric elliptic operators in divergence form whose first eigenvalue is minimal. [ABSTRACT FROM AUTHOR]

Abstract: The query complexity of the following numerical problem is studied in the quantum model of computation: consider a general elliptic partial differential equation of order in a smooth, bounded domain with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold of dimension . With the right-hand side belonging to , and the error being measured in the norm, we prove that the nth minimal quantum error is (up to logarithmic factors) of orderFor comparison, in the classical deterministic setting the nth minimal error is known to be of order , for all , while in the classical randomized setting it is (up to logarithmic factors) [Copyright &y& Elsevier]

Abstract: We study the information complexity in the randomized setting of solving a general elliptic PDE of order in a smooth, bounded domain with smooth coefficients and homogeneous boundary conditions. The solution is sought on a smooth submanifold of dimension , the right-hand side is supposed to be in , the error is measured in the norm. We show that the nth minimal error is (up to logarithmic factors) of orderFor comparison, in the deterministic setting the nth minimal error is of order for all . [Copyright &y& Elsevier]

Abstract: In this paper, the application of penalty shifting method to the calculation of optimal linewise control of the dam-detouring osmotic system for concrete dams with heterogemous is studied. The approximation program is structured, and the convergence of the method on appropriate Hilbert spaces is proved. [Copyright &y& Elsevier]

The solution of elliptic partial differential equations on composite overlapping grids using multigrid is discussed. An approach is described that provides a fast and memory efficient scheme for the solution of boundary value problems in complex geometries. The key aspects of the new scheme are an automatic coarse grid generation algorithm, an adaptive smoothing technique for adjusting residuals on different component grids, and the use of local smoothing near interpolation boundaries. Other important features include optimizations for Cartesian component grids, the use of over-relaxed red-black smoothers, and the generation of coarse grid operators through Galerkin averaging. Numerical results in two and three dimensions show that very good multigrid convergence rates can be obtained for Dirichlet, Neumann, and mixed boundary conditions. A comparison to Krylov-based solvers shows that the multigrid solver can be much faster and require significantly less memory. [ABSTRACT FROM AUTHOR]

• Developed a z-kron operation to build QTT-matrix with rows and columns in z-order. • The z-kron operation leads to O (log ⁡ n) space for QTT representations. • An algorithm to build coefficient QTT-matrix in z-order from smaller QTT-matrices. • The Algorithm has log time and space w.r.t. the number of nodes in a grid. • The coefficient QTT-matrix has log space and O (1) ranks w.r.t. the number of nodes. Elliptical PDEs are at the core of many computational problems. Sometimes it is necessary to solve them on fine meshes, which entail huge memory footprints and low computational speeds. We provide a method to solve elliptical PDEs on fine grids with lowered memory consumption and improved convergence. This paper considers one typical elliptical PDE – the Poisson equation on various polygonal domains with Dirichlet boundary conditions. The Finite Element Method (FEM) is used for numerical solution. FEM approximates a two-dimensional PDE as a system of linear equations A u = f. For an n -by- n mesh grid the sparse representation of A has the size of O (n 2). We replace the sparse matrix representation with a Quantized Tensor Train (QTT) representation to obtain O ((log ⁡ n) α) time and memory complexity to construct both A and f. This is ensured by constant-bounded QTT approximation ranks. AMEn solver is used on the final linear system, and its iterations are faster for low rank QTT approximation. To avoid rank growth caused by the intrinsic structure of A we introduce a new operation z-kron which constructs a matrix with rows and columns permuted into so called z-order. An algorithm to construct A in z-order directly in QTT with O (1) ranks w.r.t. n is provided. The proposed method is used to solve the Poisson equation on two different polygonal domains. Experiments show that our approach significantly improves memory consumption and speed over classical sparse-matrix based partial differential equation solvers like FEniCS. [ABSTRACT FROM AUTHOR]

We consider applying the preconditioned conjugate gradient (PCG) method to solving linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators with Dirichlet boundary conditions. Let (L + Σ)Σ-1(Lt + Σ) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix Σ. We propose a preconditioner M = (L + Σ)Σ-1(Lt + Σ) with block diagonal matrix Σ and lower block triangular matrix L. The diagonal blocks of Σ and the subdiagonal blocks of L are respectively the optimal sine transform approximations to the diagonal blocks of Σ and the subdiagonal blocks of L. We show that for two-dimensional domains, the construction cost of M and the cost for each iteration of the PCG algorithm are of order O(n2 log n). Furthermore, for rectangular regions, we show that the condition number of the preconditioned system M-1A is of order O(1). In contrast, the system preconditioned by the MILU and MINV methods are of order O(n). We will also show that M can be obtained from A by taking the optimal sine transform approximations of each sub-block of A. Thus, the construction of M is similar to that of Level-1 circulant preconditioners. Our numerical results on two-dimensional square and L-shaped domains show that our method converges faster than the MILU and MINV methods. Extension to higher-dimensional domains will also be discussed. © 1997 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

• A fourth-order kernel-free boundary integral method in three space dimensions. • Representation of irregular surfaces by intersection points with a Cartesian grid. • A 27-point compact finite difference scheme on irregular domains. The kernel-free boundary integral (KFBI) method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains. It evaluates boundary or volume integrals involved in the solution of boundary integral equations (BIEs) by solving equivalent but simple interface problems on regular grids, so that the integral kernel or Green function is never needed or computed. This is the essential difference of the KFBI method from the traditional ones. It takes advantage of the well-conditioning property of discrete BIEs so that the number of Krylov subspace iterations is essentially independent of discretization parameter or system dimension. This paper presents a fourth-order kernel-free boundary integral method for second-order elliptic partial differential equations on complex domains in three space dimensions, whose boundaries are given by implicitly defined surfaces. It represents the domain boundary and discretizes data on it by intersection points of the surface with Cartesian grid lines. The approach has a variety of advantages. The current work solves simple interface problems with corrected 27-point compact finite difference schemes and calculates the discrete equations with a fast Fourier transform based elliptic solver. Numerical examples show that the proposed method is efficient as well as accurate. [ABSTRACT FROM AUTHOR]

In this work, a new cubic B-spline-based semi-analytical algorithm is presented for solving 3D anisotropic convection-diffusion-reaction (CDR) problems in the inhomogeneous medium. The mathematical model is expressed by the quasi-linear second-order elliptic partial differential equations (EPDE) with mixed derivatives and variable coefficients. The final approximation is obtained as a sum of the rough primary solution and the modified spline interpolants with free parameters. The primary solution mathematically satisfies boundary conditions. Thus, the free parameters of interpolants are chosen to satisfy the governing equation in the solution domain. The numerical examples demonstrate the high accuracy of the proposed method in solving 3D CDR problems in single- and multi-connected domains. [ABSTRACT FROM AUTHOR]