Linear Trial Functions Research Articles

Multiwavelets for Second-Kind Integral Equations

We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.

A unified construction of variationalR-matrix methods: I. The Schrödinger equation

Properties of two linear integral operators and , relating function values to normal derivatives on a surface of a closed volume inside which the function satisfies the Schrodinger equation at energy E, are discussed. Variational principles for matrix elements and eigenvalues of these operators are constructed in a systematic way by using an approach of Gerjuoy et al. Some of the variational principles derived were already known in the non-relativistic R-matrix theory of scattering but others seem to be new. Applications of the Rayleigh - Ritz linear trial functions are presented.

Necessary L2-uniform convergence conditions for difference schemes for two-dimensional convection-diffusion problems

We derive necessary conditions for the uniform convergence (with respect to the perturbation parameter) in L 2 of schemes for singularly perturbed convection-diffusion problems. Applying these conditions, we show that the convergence order of the streamline diffusion finite element method with piecewise linear trial functions cannot be better than 1 2 , uniformly in the singular perturbation parameter.

A hermite CVBEM model of two-dimensional, steady-state, soil-water phase change

A linear trial function CVBEM model of a steady-state, two-dimensional freezing front is extended to the Hermite cubic polynomial trial function. The utility of this extension is demonstrated by the reduction in effort needed (over the linear trial function model) to develop an approximate boundary for error analysis, due to the derivative terms being included in the Hermite model. The modeling approach can be used not only for simple field problems, but also for the calibration of more sophisticated soil-water phase change models based on the more popular finite element and finite difference techniques.

Application of the CVBEM to multiply connected regions

The complex variable boundary element method (CVBEM) is extended to solving boundary value problems of the Laplace equation over two-dimensional multiply connected regions. Using the principal value of the Cauchy integral, linear trial functions are used to represent both the known and unknown values of the underlying analytic function. The CVBEM is then applied to developing streamlines within a river that flows past bridge piers. The purpose of the CVBEM analysis is to design the bridge pier alignment so as to minimize the disturbance to the flow field within the river.

Experimental asymptotic convergence of the collocation method for boundary integral equations on polygons governing Laplace's equation

Previously Costabel and Stephan proved the convergence of the collocation method for boundary integral equations on polygonal domains for piecewise linear trial functions which are constant on subintervals next to corners. The convergence and associated error estimates were given in suitable Sobolev spaces with appropriately weighted norms.

Consistent higher degree Petrov–Galerkin methods for the solution of the transient convection–diffusion equation

AbstractThe solution of the convection–diffusion equation for convection dominated problems is examined using both N + 1 and N + 2 degree Petrov–Galerkin finite element methods in space and a Crank–Nicolson finite difference scheme in time. While traditional N + 1 degree Petrov–Galerkin methods, which use test functions one polynomial degree higher than the trial functions, work well for steady‐state problems, they fail to adequately improve the solution for the transient problem. However, using novel N + 2 degree Petrov–Galerkin methods, which use test functions two polynomial degrees higher than the trial functions, yields dramatically improved solutions which in fact get better as the Courani number increases to 1·0. Specifically, cubic test functions with linear trial functions and quartic test functions in conjunction with quadratic trial functions are examined.Analysis and examples indicate that N + 2 degree Petrov–Galerkin methods very effectively eliminate space and especially time truncation errors. This results in substantially improved phase behaviour while not adversely affecting the ratio of numerical to analytical damping.

A numerical study of nonlinear waves arising in a one-dimensional model of a fluidized bed

A third-order nonlinear problem, which under certain circumstances approximates the flow in a 1-dimensional fluidized bed, is solved numerically. A Petrov-Galerkin finite element method, with piecewise linear trial functions and cubic spline test functions, is used for the discretization in space. If product approximation is applied to the numerical representation of the nonlinear terms, then the scheme is fourth-order in space in the finite difference sense. Second-order backward differentiation is used for the time stepping. It is found that, while certain values of the time step may produce stable results, a reduction in the time step introduces instability. This is confirmed by a von Neumann stability analysis of a simplified case and is also shown to be reasonable in view of the continuous problem which contains stable and unstable modes. A set of numerical experiments is presented.

Extension of the CVBEM to higher-order trial functions

The higher-order trial functions, such as the parabolic, cubic, and Hermite cubic polynomial functions, for the complex boundary element method are derived and their computer programs are developed. Using the considered higher-order trial function, models obtained compatible results to the linear trial function model.

On the convergence of collocation methods for Symm’s integral equation on open curves

Recently, Costabel and Stephan in [8] presented convergence proofs for collocation with piecewise linear trial functions for Symm’s integral equation on plane closed curves with corners. In this article we prove the convergence of the above collocation method in the case of open curves. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes. Numerical experiments based on the implementation of [6] show experimental orders of convergence which confirm our theoretical results on the asymptotic rates of convergence.

On the Convergence of Collocation Methods for Boundary Integral Equations on Polygons

The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind [4], [7], one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves [2], [3], [17], [26]—[29], and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains [5], [8], [9]. Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann’s integral equation and Symm’s integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes.

Error estimates for the finite element solution of quasilinear obstacle problems*

This paper focusses on quasilinear second order differential operators. Concerning their strong monotonicity and Lipschitz continuity sufficient conditions are stated which are both explicit and easy to show. Applying these functional analytic properties to nonlinear obstacle problems we prove error estimates in energy norm for the finite element method with linear trial functions.

On the convergence of collocation methods for boundary integral equations on polygons

The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind [4], [7], one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves [2], [3], [17], [26]—[29], and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains [5], [8], [9]. Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann’s integral equation and Symm’s integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes.

A remark on theL ? bounds of the Ritz operator associated with a finite element approximation

For the elliptic operatorA of second order, its finite element approximationA h is considered by piecewise linear trial functions. AnL ∞ bound on the Ritz operator is shown by Stampacchia's method, which implies a discrete elliptic-Sobolev inequality forA h.

Analysis of a Fully Discrete Scheme for Neutron Transport in Two-Dimensional Geometry

We derive error estimates for a fully discrete scheme for the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry obtained by using a special quadrature rule for the angular variable and the discontinuous Galerkin finite element method with piecewise linear trial function for the space variable.

A finite element method for a singularly perturbed boundary value problem

We examine the problem:eu″+a(x)u′−b(x)u=f(x) for 0 0,b(x)>β,α 2 = 4eβ>0,a, b andf inC 2 [0, 1], e in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh 2, whereC is independent ofh and e). With a natural choice of trial functions, uniform first order accuracy is obtained in theL ∞ (0, 1) norm. On choosing piecewise linear trial functions (“hat” functions), uniform first order accuracy is obtained in theL 1 (0, 1) norm.

A UNIFIED MODEL OF RADIALLY SYMMETRIC HEAT CONDUCTION

The nodal domain integration method is applied to a radially symmetric heat conduction problem where the solution domain is discretized into irregular radial finite elements, and the state variable is approximated by a spatial linear trial function within each finite element. The resulting finite-element model represents the well-known Galerkin finite-element, subdomain-integration, and an integrated finite-difference numerical statement as well as an infinity of other mass-lumped matrix schemes. From the NDI approach, the several numerical modeling techniques are unified into one global domain model where each submodel can be obtained by the specification of a single mass-lumping parameter.

A spline collocation method for singular integral equations with piecewise continuous coefficients

We consider the collocation method with piecewise linear trial functions for systems of singular integral equations with Cauchy kernel and piecewise continuous coefficients. Necessary and sufficient conditions for the stability in L2 are given. The results are obtained in the case of a closed Ljapunov curve as well as in the case of an interval. The proof of the main theorem is based on a modification of the Banach algebra technique established in the local principle by Gohberg and Krupnik [2]. Our results extend those obtained by Prosdorf and Schmidt [9, 10] from the case of continuous coefficients and unit circle to the case of piecewise continuous coefficients.

Nodal domain integration model of two-dimensional advection-diffusion processes

The nodal domain integration method is applied to a two-dimensional advection-diffusion process in an anisotropic inhomogeneous medium. The domain is discretised into the union of irregular triangle finite elements with vertex-located nodal points and a linear trial function is used to approximate the governing flow equation's state variable in each element. Non-linear parameters are assumed quasi-constant for small durations in time in each element. The resulting numerical model represents the Galerkin and subdomain integration weighted residual methods and the integrated finite difference method as special cases. Both Dirichlet and Neumann boundary conditions are accommodated in a manner similar to the Galerkin finite element approach.

MASS-LUMPING NUMERICAL MODELS OF THREE-DIMENSIONAL HEAT CONDUCTION

Abstract A variable mass-lumping numerical model (nodal domain integration) of three-dimensional heat conduction in an inhomogeneous continuum is developed The domain is discretized by tetrahedron-shaped elements and the state variable is approximated by linear trial functions. The resulting model represents the Galerkin finite-element, subdomain intergration, and integrated finite-difference methods as special cases and accommodates both Dirichlet and Neumann boundary conditions similar to a Galerkin finite-element model Consequently, a unified domain numerical model is developed that readily represents each of the abovementioned domain numerical methods and an infinity of finite-element mass-lumping schemes by the specification of a single constant model parameter. Application of the nodal domain integration model to linear heat conduction problems indicates that the degree of model mass lumping must vary to minimize the approximation error.