# 统计代写|主成分分析代写Principal Component Analysis代考|Finite Element Method: Discretized Solution for PDEs

The passage describes how the method of weighted residuals, specifically Galerkin’s method, is adapted to the finite element method for solving partial differential equations (PDEs) in a discretized manner. The main idea is to divide the problem domain into smaller elements and express the approximate solution within each element as a linear combination of local trial functions that interpolate the solution at nodal points.

For the given PDE:

d2ydx2+f(x)=0a≤x≤b\frac{d^2y}{dx^2} + f(x) = 0 \quad a \leq x \leq b dx 2

d 2 y ​ +f(x)=0a≤x≤b with boundary conditions y(a)=yay(a) = y_ay(a)=y a ​ and y(b)=yby(b) = y_by(b)=y b ​ , the domain is broken down into MMM elements. Over each element, a piecewise-linear trial function is defined such that it is nonzero only in its respective interval and passes through the nodal points where the solution is known or unknown yet to be determined.

The approximate solution y∗(x)y^*(x)y ∗ (x) is constructed using these local trial functions, and the residual R(x;yi)R(x; y_i)R(x;y i ​ ) is computed by substituting y∗(x)y^*(x)y ∗ (x) into the governing equation. Applying Galerkin’s method, the integral of the product of each trial function and the residual over the global domain is set to zero, which leads to a system of algebraic equations. These equations are assembled into a matrix form [K]{y}={F}[K]\{y\}=\{F\}[K]{y}={F}, where KKK is the stiffness matrix, {y}\{y\}{y} is the vector of nodal displacements (solution values at nodes), and {F}\{F\}{F} is the vector of nodal forces (resultants of the applied load and boundary conditions).

To derive the element-level equations, the problem is considered locally within each finite element with its own boundary conditions. For instance, for the problem:

xd2ydx2+dydx−4x=01≤x≤2x \frac{d^2y}{dx^2} + \frac{dy}{dx} – 4x = 0 \quad 1 \leq x \leq 2x dx 2

d 2 y ​ + dx dy ​ −4x=01≤x≤2 with boundary conditions y(1)=y(2)=0y(1) = y(2) = 0y(1)=y(2)=0, a two-node linear element is employed with trial functions N1(x)N_1(x)N 1 ​ (x) and N2(x)N_2(x)N 2 ​ (x). The element residual is integrated over the element length, applying integration by parts to reduce the highest derivative and introduce the gradient boundary conditions.

For this specific problem, the element residual equation becomes, after integration by parts:

∫x1x2Ni(xd2yedx2+dyedx−4x)dx=0i=1,2\int_{x_1}^{x_2} N_i \left(x \frac{d^2y_e}{dx^2} + \frac{dy_e}{dx} – 4x \right) dx = 0 \quad i = 1, 2∫ x 1 ​

x 2 ​

​ N i ​ (x dx 2

d 2 y e ​

​ + dx dy e ​

​ −4x)dx=0i=1,2

Integrating the first term by parts and applying the boundary conditions, we get a set of element stiffness matrix entries kijk_{ij}k ij ​ and element nodal forces FiF_iF i ​ . When the element equations are assembled across the entire domain, the global system is formed, ensuring continuity of the solution and its derivative at the nodes shared by adjacent elements.

However, the exact solution for the provided differential equation is already given as:

y(x)=x2−3ln⁡2ln⁡x−1y(x) = x^2 – \frac{3 \ln 2}{\ln x – 1}y(x)=x 2 − lnx−1 3ln2 ​

For the finite element solution using a two-node element, the element solution is approximated as a linear combination of the shape functions N1(x)N_1(x)N 1 ​ (x) and N2(x)N_2(x)N 2 ​ (x) multiplied by the unknown nodal values y1y_1y 1 ​ and y2y_2y 2 ​ , respectively. The residual equation is formulated accordingly and integrated to produce the local element stiffness matrix and force vector components. Once assembled, these local contributions create the overall system of equations that matches the global behavior of the continuous problem up to the degree of approximation inherent in the finite element method. The accuracy of the finite element solution can be assessed by examining the gradient discontinuities at nodes or inter-element boundaries.

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