scholarly journals Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Liqun Wang ◽  
Liwei Shi
Keyword(s):  
Finite Element ◽  
Elliptic Equation ◽  
Numerical Method ◽  
Piecewise Linear ◽  
Second Order ◽  
Three Dimensions ◽  
Matrix Coefficient ◽  
Jump Conditions ◽  
Irregular Domains ◽  
Nonsmooth Boundary

We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the L∞ norm in both two and three dimensions and numerically very stable.

HIGH-ORDER GALERKIN APPROXIMATIONS FOR PARAMETRIC SECOND-ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

2013 ◽  
Vol 23 (09) ◽  
pp. 1729-1760 ◽  
Author(s):  
VICTOR NISTOR ◽  
CHRISTOPH SCHWAB
Keyword(s):  
Finite Element ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Regularity Result ◽  
Second Order ◽  
Three Dimensions ◽  
Open Unit ◽  
Partial Differential ◽  
Open Unit Ball ◽  
Galerkin Approximations

Let D ⊂ ℝd, d = 2, 3, be a bounded domain with piecewise smooth boundary, Y = ℓ∞(ℕ) and U = B1(Y), the open unit ball of Y. We consider a parametric family (Py)y∈U of uniformly strongly elliptic, second-order partial differential operators Py on D. Under suitable assumptions on the coefficients, we establish a regularity result for the solution u of the parametric boundary value problem Py u(x, y) = f(x, y), x ∈ D, y ∈ U, with mixed Dirichlet–Neumann boundary conditions on ∂d D and, respectively, on ∂n D. Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces [Formula: see text] of Kondrat'ev type. We prove that the (Py)y ∈ U admit a shift theorem that is uniform in the parameter y ∈ U. Specifically, if the coefficients of P satisfy [Formula: see text], y = (yk)k≥1 ∈ U and if the sequences [Formula: see text] are p-summable in k, for 0 < p< 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials Lν(y) such that the corresponding sequence [Formula: see text], where [Formula: see text]. We also show optimal algebraic orders of convergence for the Galerkin approximations uℓ of the solution u using suitable Finite Element spaces in two and three dimensions. Namely, let t = m/d and s = 1/p-1/2, where [Formula: see text], 0 < p < 1. We show that, for each m ∈ ℕ, there exists a sequence {Sℓ}ℓ≥0 of nested, finite-dimensional spaces Sℓ ⊂ L2(U;V) such that the Galerkin projections uℓ ∈ Sℓ of u satisfy ‖u - uℓ‖L2(U;V) ≤ C dim (Sℓ)- min {s, t} ‖f‖Hm-1(D), dim (Sℓ) → ∞. The sequence Sℓ is constructed using a sequence Vμ⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace Sℓ is defined by a finite subset [Formula: see text] of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λℓ in the Legendre chaos expansion of u which are approximated by vν ∈ Vμ(ℓ, ν), for each ν ∈ Λℓ, with a suitable choice of μ(ℓ, ν).

scholarly journals A Numerical Method for Solving 3D Elasticity Equations with Sharp-Edged Interfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Liqun Wang ◽  
Songming Hou ◽  
Liwei Shi
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
Main Idea ◽  
Three Dimensions ◽  
Test Function ◽  
Linear System Of Equations ◽  
Elasticity Equations ◽  
Matrix Coefficients ◽  
The Matrix ◽  
3D Elasticity

Interface problems occur frequently when two or more materials meet. Solving elasticity equations with sharp-edged interfaces in three dimensions is a very complicated and challenging problem for most existing methods. There are several difficulties: the coupled elliptic system, the matrix coefficients, the sharp-edged interface, and three dimensions. An accurate and efficient method is desired. In this paper, an efficient nontraditional finite element method with nonbody-fitting grids is proposed to solve elasticity equations with sharp-edged interfaces in three dimensions. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up).

scholarly journals A Numerical Method for Solving Elasticity Equations with Interfaces

2012 ◽  
Vol 12 (2) ◽  
pp. 595-612 ◽  
Author(s):  
Songming Hou ◽  
Zhilin Li ◽  
Liqun Wang ◽  
Wei Wang
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
Main Idea ◽  
Jump Conditions ◽  
Challenging Problem ◽  
Smooth Solutions ◽  
Order Accuracy ◽  
Test Function ◽  
Linear System Of Equations ◽  
Elasticity Equations

AbstractSolving elasticity equations with interfaces is a challenging problem for most existing methods. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner.

Superconvergence Recovery for the Gradient of the Trilinear Finite Element

2011 ◽  
Vol 268-270 ◽  
pp. 1021-1024
Author(s):  
Jing Hong Liu ◽  
De Cheng Yin
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Three Dimensions ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Interpolation Postprocessing ◽  
Linear Elements

For a second-order elliptic boundary value problem in three dimensions, we use an interpolation postprocessing technique to obtain recovered gradients of tri- linear elements over regular meshes. Further, superconvergence of these gradients are proved.

A Numerical Method for Solving Matrix Coefficient Heat Equations with Interfaces

2015 ◽  
Vol 8 (4) ◽  
pp. 475-495
Author(s):  
Liqun Wang ◽  
Liwei Shi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Euler Method ◽  
Matrix Coefficient ◽  
Heat Equations ◽  
Second Derivative ◽  
Order Accuracy

AbstractIn this paper, we propose a numerical method for solving the heat equations with interfaces. This method uses the non-traditional finite element method together with finite difference method to get solutions with second-order accuracy. It is capable of dealing with matrix coefficient involving time, and the interfaces under consideration are sharp-edged interfaces instead of smooth interfaces. Modified Euler Method is employed to ensure the accuracy in time. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner. Extensive numerical experiments illustrate the feasibility of the method.

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Xiaofei Guan ◽  
Mingxia Li ◽  
Wenming He ◽  
Zhengwu Jiang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elliptic Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
High Degree ◽  
Superconvergence Results ◽  
Element Method

AbstractIn this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x 0 for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h p+1 |ln h|2) and O(h p+2 |ln h|2) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.

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