An electric-analog simulation of elliptic partial differential equations using finite element theory

Elliptic partial differential equations can be solved using the Galerkin-finite element method to generate the approximating algebraic equations, and an electrical network to solve the resulting matrices. Some element configurations require the use of networks containing negative resistances which, while physically realizable, are more expensive and time-consuming to construct.     

Finite element approximation of some indefinite elliptic problems

We consider the finite element approximation of some indefinite Neumann problems in a domain of IR^N. From the Fredholm Alternative this kind of problem admits a solution if and only if the right hand term has zero mean value with respect to a measure whose density m is the solution of a homogeneous adjoint problem. The first step consists in the construction of piecewise linear finite element approximations m_h of m, showing their optimal rate of convergence both in energy and L^p norms. The functions m_h are then shown to be crucial in testing admissible data for the Neumann problem and also in its numerical resolution (actually, the standard Galerkin approximation may not be solvable without suitable perturbations of the data).     

A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations

We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second-order coercive linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic ‘energy’ reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine ‘truth-mesh’ discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid-flux candidate Lagrange multipliers generated by a K-element coarser ‘working-mesh’ approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomain-local, symmetric Neumann problems. The technique is illustrated for the convection-diffusion and linear elasticity equations.     

On solving elliptic stochastic partial differential equations

A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.     

A note on the design of hp-adaptive finite element methods for elliptic partial differential equations

We introduce an hp-adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp-extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local Legendre series expansion. Numerical experiments confirm the robustness and reliability of the proposed algorithm.     

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.     

A complex variable boundary element method for elliptic partial differential equations in a multiple-connected region

A simple boundary element method based on the Cauchy integral formulae is proposed for the numerical solution of a class of boundary value problems involving a system of elliptic partial differential equations in a multiple-connected region of infinite extent. It can be easily and efficiently implemented on the computer.     

Finite difference solutions of dispersive partial differential equations

The second order leapfrog method is used to discretize the linearized KdV equation which is itself a dispersive partial differential equation. The resulting difference equation is solved and analyzed in terms of its dispersion relation and propagation properties. Numerical experiments are included to illustrate some of the results obtained. Finally, a novel aliasing error which affects the propagation of wave packets is presented     

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.     

Explicit group over-relaxation methods for solving elliptic partial differential equations

Previous block (or line) iterative methods have been implicit in nature where a group of equations (or points on the grid mesh) are treated implicitly [2] and solved directly by a specialised algorithm, this has become the standard technique for solving the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques.The aim of this paper is to show that if a small group of points (i.e. 2, 4, 9, 16 or 25 point group) is chosen then each group can easily be initially inverted leading to a new class of Group Explicit iterative methods. A comparison with the usual 1-line and 2-line block S.O.R. schemes for the model problem confirm the new techniques to be computationally superior.     

Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with (approximately) affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N, typically very small, and the (approximate) parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.In our earlier work, we develop a rigorous a posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not complete. In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interest and associated rigorous a posteriori error bounds; and the latter incorporate both numerical discretization and model truncation effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

Development,evaluation and selection of methods for elliptic partial differential equations

We present a framework within which to evaluate and compare computational methods to solve elliptic partial differential equations. We then report on the results of comparisons of some classical methods as well as a new one presented here. Our main motivation is the belief that the standard finite difference methods are almost always inferior for solving elliptic problems and our results are strong evidence that this is true. The superior methods are higher order (fourth or more instead of second) and we describe a new collocation finite element method which we believe is more efficient and flexible than the other well known methods, e.g., fourth order finite differences, fourth order finite element methods of Galerkin, Rayleigh-Ritz or least squares type.Our comparisons are in the context of the relatively complicated problems that arise in realistic applications. Our conclusion does not hold for simple model problems (e.g., Laplaces equation on a rectangle) where very specialized methods are superior to the generally applicable methods that we consider. The accurate and relatively simple treatment of boundary conditions involving curves and derivations is a feature of our collocation method.     

A direct approximation method for parameter estimation in partial differential equations

We investigate a general method for estimating numerical values for the parameters which occur in certain partial differential equations. The method uses direct approximations of the solution and selects parameters to match the differential equation. Several numerical examples are presented to illustrate the technique and compare it with initial-boundary value methods.     

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

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.     

Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary

The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems.In [1], we have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as auto. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature.In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.     

The solution of elliptic partial differential equations by a new block over-relaxation technique

In this paper, a new explicit 4-pint block over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the discretization of self-adjoint elliptic partial differential equations. A comparison with the implicit line and 2-line block SOR schemes for the model problem shows the new technique to be competitive     

The role of the multiquadric shape parameters in solving elliptic partial differential equations

This study examines the generalized multiquadrics (MQ), φ_j(x) = [(x−x_j)²+c_j²]^β in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent β as well as c_j² can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of β as well as c_j² where c_j² can be different over the interior and on the boundary. The results show that increasing ,β has the most important effect on convergence, followed next by distinct sets of (c_j²)Ω∂Ω ≪ (c_j²)∂Ω. Additional convergence accelerations were obtained by permitting both (c_j²)Ω∂Ω and (c_j²)∂Ω to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.