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:''For the stiffness tensor in solid mechanics, see Hooke's law#Matrix representation (stiffness tensor).'' In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. ==The stiffness matrix for the Poisson problem== For simplicity, we will first consider the Poisson problem : on some domain Ω, subject to the boundary condition ''u'' = 0 on the boundary of Ω. To discretize this equation by the finite element method, one chooses a set of ''basis functions'' defined on Ω which also vanish on the boundary. One then approximates : The coefficients ''u''1, ..., ''u''''n'' are determined so that the error in the approximation is orthogonal to each basis function ''φ''''i'': : The stiffness matrix is the n-element square matrix A defined by : By defining the vector ''F'' with components ''F''''i'' = (''φ''''i'', ''f''), the coefficients ''u''''i'' are determined by the linear system ''AU'' = ''F''. The stiffness matrix is symmetric, i.e. ''A''''ij'' = ''A''''ji'', so all its eigenvalues are real. Moreover, it is a strictly positive-definite matrix, so that the system ''AU'' = ''F'' always has a unique solution. (For other problems, these nice properties will be lost.) Note that the stiffness matrix will be different depending on the computational grid used for the domain and what type of finite element is used. For example, the stiffness matrix when piecewise quadratic finite elements are used will have more degrees of freedom than piecewise linear elements. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「stiffness matrix」の詳細全文を読む スポンサード リンク
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