Computing H1 norm numerically












1












$begingroup$


I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm







share|cite|improve this question









$endgroup$












  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05
















1












$begingroup$


I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm







share|cite|improve this question









$endgroup$












  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05














1












1








1


1



$begingroup$


I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm







share|cite|improve this question









$endgroup$




I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm




numerical-methods norm






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Feb 21 '14 at 3:24









RambiRambi

185




185












  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05


















  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05
















$begingroup$
$u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
$endgroup$
– Erick Wong
Feb 21 '14 at 3:40




$begingroup$
$u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
$endgroup$
– Erick Wong
Feb 21 '14 at 3:40












$begingroup$
haha...oops how would you actually compute that in matlab?
$endgroup$
– Rambi
Feb 21 '14 at 4:05




$begingroup$
haha...oops how would you actually compute that in matlab?
$endgroup$
– Rambi
Feb 21 '14 at 4:05










1 Answer
1






active

oldest

votes


















0












$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f684389%2fcomputing-h1-norm-numerically%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43
















0












$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43














0












0








0





$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$



The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Feb 21 '14 at 13:07









Algebraic PavelAlgebraic Pavel

16.4k31840




16.4k31840












  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43


















  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43
















$begingroup$
sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
$endgroup$
– Rambi
Feb 21 '14 at 14:34




$begingroup$
sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
$endgroup$
– Rambi
Feb 21 '14 at 14:34












$begingroup$
I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
$endgroup$
– Algebraic Pavel
Feb 21 '14 at 14:43




$begingroup$
I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
$endgroup$
– Algebraic Pavel
Feb 21 '14 at 14:43


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f684389%2fcomputing-h1-norm-numerically%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Bundesstraße 106

Image
Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück
Read more

Verónica Boquete

Image
Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl
Read more

Ida-Boy-Ed-Garten

Image
Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
Read more