https://blog.matnoble.top/math/fem/MatNoble 个人技术博客，聚焦大学数学可视化、算法思维与数据结构、LaTeX 科技论文与 Beamer 课件排版，以及大数据与分布式系统实践。记录从程序员到高校数学教师的深度技术思考与学术沉淀。Hugo 0.165.0 & FixIt v0.4.5zh-CNme@matnoble.top (MatNoble)me@matnoble.top (MatNoble)Tue, 08 Sep 2026 13:00:21 +0800https://blog.matnoble.top/math/fem/finite-element-basis-functions/Mon, 30 Dec 2019 00:00:00 +0800me@matnoble.top (MatNoble)https://blog.matnoble.top/math/fem/finite-element-basis-functions/MATH简述有限元<figure class="caption-wrapper"> <a class="lightgallery" href="https://imgkr.cn-bj.ufileos.com/64498351-25f9-435e-8951-704f5883d9e0.png" title="简述有限元: 逼近函数 IV" data-thumbnail="https://imgkr.cn-bj.ufileos.com/64498351-25f9-435e-8951-704f5883d9e0.png" data-sub-html="<h2>简述有限元: 逼近函数 IV</h2>"> <img class="caption" src="https://imgkr.cn-bj.ufileos.com/64498351-25f9-435e-8951-704f5883d9e0.png" title="简述有限元: 逼近函数 IV" alt="简述有限元: 逼近函数 IV" style="max-width: 100%; width: 95%; margin: 0 auto; display: block;" loading="lazy"> </a><figcaption class="image-caption"> <span>简述有限元: 逼近函数 IV</span> </figcaption></figure> <h2 class="heading-element" id="上期回顾"><span>上期回顾</span> <a href="#%e4%b8%8a%e6%9c%9f%e5%9b%9e%e9%a1%be" class="heading-mark"> <svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg> </a> </h2><h3 class="heading-element" id="程序"><span>程序</span> <a href="#%e7%a8%8b%e5%ba%8f" class="heading-mark"> <svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg> </a> </h3><blockquote> <p>例题 1<br> 利用拉格朗日多项式做基函数编程实现: 设函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>10</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f(x)=10(x-1)^2-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">10</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, 在线性函数空间中找到最佳逼近函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>, 求解域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Omega = [1, 2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">]</span></span></span></span>, 两个真解点为: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo separator="true">,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">x_0 = 1+1/3, x_1 = 1+2/3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2/3</span></span></span></span>.</p>https://blog.matnoble.top/math/fem/function-approximation-interpolation/Sat, 28 Dec 2019 00:00:00 +0800me@matnoble.top (MatNoble)https://blog.matnoble.top/math/fem/function-approximation-interpolation/MATH简述有限元<figure class="caption-wrapper"> <a class="lightgallery" href="https://imgkr.cn-bj.ufileos.com/d6b2385b-358d-42a3-9d01-5eef5d9ab76c.jpeg" title="简述有限元: 逼近函数 III" data-thumbnail="https://imgkr.cn-bj.ufileos.com/d6b2385b-358d-42a3-9d01-5eef5d9ab76c.jpeg" data-sub-html="<h2>简述有限元: 逼近函数 III</h2>"> <img class="caption" src="https://imgkr.cn-bj.ufileos.com/d6b2385b-358d-42a3-9d01-5eef5d9ab76c.jpeg" title="简述有限元: 逼近函数 III" alt="简述有限元: 逼近函数 III" style="max-width: 100%; width: 95%; margin: 0 auto; display: block;" loading="lazy"> </a><figcaption class="image-caption"> <span>简述有限元: 逼近函数 III</span> </figcaption></figure> <h2 class="heading-element" id="前情回顾"><span>前情回顾</span> <a href="#%e5%89%8d%e6%83%85%e5%9b%9e%e9%a1%be" class="heading-mark"> <svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg> </a> </h2><h3 class="heading-element" id="引子"><span>引子</span> <a href="#%e5%bc%95%e5%ad%90" class="heading-mark"> <svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg> </a> </h3><p>我们用<a href="https://blog.matnoble.top/math/fem/function-approximation-intro/">两次推送</a>, 讨论了用<a href="https://zh.wikipedia.org/wiki/%e6%9c%80%e5%b0%8f%e4%ba%8c%e4%b9%98%e6%b3%95" title="最小二乘法" target="_blank" rel="external nofollow noopener noreferrer">最小二乘法</a>和<a href="https://zh.wikipedia.org/wiki/%e4%bc%bd%e8%be%bd%e9%87%91%e6%b3%95" title="伽辽金法" target="_blank" rel="external nofollow noopener noreferrer">伽辽金法</a>逼近函数, 二者出发点不同, 但最终都是化无限维为有限维问题来求解, 即将寻找最优逼近函数问题化为解线性方程组.</p>https://blog.matnoble.top/math/fem/function-approximation-orthogonal-basis/Thu, 26 Dec 2019 00:00:00 +0800me@matnoble.top (MatNoble)https://blog.matnoble.top/math/fem/function-approximation-orthogonal-basis/MATH简述有限元<figure class="caption-wrapper"> <a class="lightgallery" href="https://imgkr.cn-bj.ufileos.com/62f007ea-3d01-45a5-8ec7-4bfe52d009ed.jpeg" title="简述有限元: 逼近函数 II" data-thumbnail="https://imgkr.cn-bj.ufileos.com/62f007ea-3d01-45a5-8ec7-4bfe52d009ed.jpeg" data-sub-html="<h2>简述有限元: 逼近函数 II</h2>"> <img class="caption" src="https://imgkr.cn-bj.ufileos.com/62f007ea-3d01-45a5-8ec7-4bfe52d009ed.jpeg" title="简述有限元: 逼近函数 II" alt="简述有限元: 逼近函数 II" style="max-width: 100%; width: 95%; margin: 0 auto; display: block;" loading="lazy"> </a><figcaption class="image-caption"> <span>简述有限元: 逼近函数 II</span> </figcaption></figure>https://blog.matnoble.top/math/fem/function-approximation-intro/Tue, 24 Dec 2019 00:00:00 +0800me@matnoble.top (MatNoble)https://blog.matnoble.top/math/fem/function-approximation-intro/MATH简述有限元<figure class="caption-wrapper"> <a class="lightgallery" href="https://imgkr.cn-bj.ufileos.com/2183d598-1f26-450f-b94d-daabdfab77d6.jpg" title="简述有限元: 逼近函数 I" data-thumbnail="https://imgkr.cn-bj.ufileos.com/2183d598-1f26-450f-b94d-daabdfab77d6.jpg" data-sub-html="<h2>简述有限元: 逼近函数 I</h2>"> <img class="caption" src="https://imgkr.cn-bj.ufileos.com/2183d598-1f26-450f-b94d-daabdfab77d6.jpg" title="简述有限元: 逼近函数 I" alt="简述有限元: 逼近函数 I" style="max-width: 100%; width: 95%; margin: 0 auto; display: block;" loading="lazy"> </a><figcaption class="image-caption"> <span>简述有限元: 逼近函数 I</span> </figcaption></figure>https://blog.matnoble.top/math/fem/vector-approximation/Thu, 12 Dec 2019 00:00:00 +0800me@matnoble.top (MatNoble)https://blog.matnoble.top/math/fem/vector-approximation/MATH简述有限元<figure class="caption-wrapper"> <a class="lightgallery" href="https://cdn.jsdelivr.net/gh/MatNoble/Images@master/20260104202457230.png" title="简述有限元：逼近向量" data-thumbnail="https://cdn.jsdelivr.net/gh/MatNoble/Images@master/20260104202457230.png" data-sub-html="<h2>简述有限元：逼近向量</h2>"> <img class="caption" src="https://cdn.jsdelivr.net/gh/MatNoble/Images@master/20260104202457230.png" title="简述有限元：逼近向量" alt="简述有限元：逼近向量" style="max-width: 100%; width: 95%; margin: 0 auto; display: block;" loading="lazy"> </a><figcaption class="image-caption"> <span>简述有限元：逼近向量</span> </figcaption></figure> <p><a href="https://en.wikipedia.org/wiki/Finite_element_method" title="有限元" target="_blank" rel="external nofollow noopener noreferrer">有限元</a> <strong>博大精深</strong>, 开设本专题意在督促自己学习, 只能讲述一些最简单的有限元知识. 本主题将注重<strong>数学推导</strong>和有关<strong>程序实现</strong>(Matlab 为主, Python 为辅), 配以适当图示, 尽量做到<strong>不晦涩</strong>, 强调<strong>趣味性</strong>和<strong>易读性</strong>.</p>