An Accurate High-order Method to Solve the Helmholtz Boundary Value Problem for the 3d Laplace Equation
نویسندگان
چکیده
The 3D Laplace equation is one of the important PDEs of physics and describes the phenomonology of electrostatics and magnetostatics. Frequently very precise solution of this PDE is required; but with conventional finite element or finite difference codes this is difficult to achieve because of the need for an exceedingly fine mesh which leads to often prohibitive CPU time. We present an alternate approach based on high-order quadrature and a high-order finite element method. Both of the ingredients become possible via the use of high-order differential algebraic methods. Various examples of the method and the precision that can be achieved will be given. For example, using only about 100 finite elements of order 7, accuracies in the range of 10−6 can be obtained in the 3D case. AMS Subject Classification: 33F99, 35J05, 65N99
منابع مشابه
An efficient method for the numerical solution of Helmholtz type general two point boundary value problems in ODEs
In this article, we propose and analyze a computational method for numerical solution of general two point boundary value problems. Method is tested on problems to ensure the computational eciency. We have compared numerical results with results obtained by other method in literature. We conclude that propose method is computationally ecient and eective.
متن کاملOn the Numerical Solution of One Dimensional Schrodinger Equation with Boundary Conditions Involving Fractional Differential Operators
In this paper we study of collocation method with Radial Basis Function to solve one dimensional time dependent Schrodinger equation in an unbounded domain. To this end, we introduce artificial boundaries and reduce the original problem to an initial boundary value problem in a bounded domain with transparent boundary conditions that involves half order fractional derivative in t. Then in three...
متن کاملApplication of Decoupled Scaled Boundary Finite Element Method to Solve Eigenvalue Helmholtz Problems (Research Note)
A novel element with arbitrary domain shape by using decoupled scaled boundary finite element (DSBFEM) is proposed for eigenvalue analysis of 2D vibrating rods with different boundary conditions. Within the proposed element scheme, the mode shapes of vibrating rods with variable boundary conditions are modelled and results are plotted. All possible conditions for the rods ends are incorporated ...
متن کاملFinite Element Analysis of a One-dimensional Helmholtz Equation
Many science and engineering problems can be mathematically modeled as a boundary value problem. Thus, it is very important to build a method to solve for a boundary value problem. For some simple boundary value problems, for example, a two-dimensional Laplace equation, it may be possible to solve it analytically by separation of variables. However, in most applications, boundary value problems...
متن کاملCompare Adomian Decomposition Method and Laplace Decomposition Method for Burger's-Huxley and Burger's-Fisher equations
In this paper, Adomian decomposition method (ADM) and Laplace decomposition method (LDM) used to obtain series solutions of Burgers-Huxley and Burgers-Fisher Equations. In ADM the algorithm is illustrated by studying an initial value problem and LDM is based on the application of Laplace transform to nonlinear partial differential equations. In ADM only few terms of the expansion are required t...
متن کامل