Graph of a matrix for a 2D grid using certain discretization scheme

I have a 2D grid which will result in a sparse matrix when discretized with 2nd order central difference scheme using finite difference method (5 point stencil). Is there a way to generate a matrix with fill-ins that shows entry for respective position of the grid point.

(This question is aimed at those with mathematics background, specifically those who understand the sparse linear systems and discretization schemes )

example image of the matrix :

Note : above image is not the matrix structure that will result from the discretization of the grid below but I am looking at a image similar to above for the discretization of my grid using 2nd order central difference scheme (for details one can check out this page)

Code for my grid points

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
[darkstyle/.style={circle,draw,minimum size=25}]
\foreach \x in {0,...,4}
\foreach \y in {0,...,4}
{\pgfmathtruncatemacro{\label}{\x - 5 *  \y +21}
\node [darkstyle]  (\x\y) at (2.5*\x,2.5*\y) {\label};}

\foreach \x in {0,...,4}
\foreach \y [count=\yi] in {0,...,3}
\draw (\x\y)--(\x\yi) (\y\x)--(\yi\x) ;
\end{tikzpicture}
\end{document}
%resulting grid shown below


The solution I came up with doesn't link the grid to my discretization scheme and generate the matrix gaph (which seems a bit complicated task of assembling something called stiffness matrix, I'm asking too much from TEX :P) nevertheeless with a very bad piece of code, which I don't like, I managed to generate the graph I'm looking for. Here it is:

\begin{tikzpicture}
%draws the outer box
\draw (0.15,-0.05) rectangle (-3.8,3.9);

\foreach \x in  {0,...,25}
\foreach \y in  {0,...,25}
\foreach \position in {(0.15*-\x,0.15*\y)}
{\ifnum \x=\y
{ \draw[fill=black] \position rectangle +(0.1,0.1);}
\fi}

\foreach \x in  {1,...,25}
\foreach \y in  {1,...,25}
\foreach \position in {(0.15 + 0.15*-\x,0.15*\y)}
{\ifnum \x = \y
{\draw[fill=black] \position rectangle +(0.1,0.1);}
\fi}

\foreach \x in  {0,...,25}
\foreach \y in  {0,...,24}
\foreach \position in {( -0.15+0.15*-\x,0.15*\y)}
{\ifnum \x = \y
{ \draw[fill=black] \position rectangle +(0.1,0.1);}
\fi}

\foreach \x in  {5,...,25}
\foreach \y in  {0,...,25}
\foreach \position in {( 0.75+0.15*-\x,0.15*\y)}
{\ifnum \x = \y
{\draw[fill=black] \position rectangle +(0.1,0.1);}
\fi}

\foreach \x in  {0,...,20}
\foreach \y in  {0,...,25}
\foreach \position in {( -0.75+0.15*-\x,0.15*\y)}
{\ifnum \x = \y
{\draw[fill=black] \position rectangle +(0.1,0.1);}
\fi}
\end{tikzpicture}


The graph of the matrix looks like this :

But I'll be happy if someone can give me suggestions to clean up the code :)

OK, thanks for your answer, now I start to understand what you want. Here is a somewhat simpler code.

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{plotmarks}
\begin{document}
\pgfsetplotmarksize{0.05cm}
\begin{tikzpicture}
%draws the outer box
\draw (0.15,-0.15) rectangle (-3.95,3.9);

\foreach \x in  {0,...,25}
{\foreach \y in  {0,...,25}
{\pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)<2,0,1)}
\ifnum\Z=0
\node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
\else
\pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)==4,0,1)}
\ifnum\Z=0
\node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
\fi
\fi}
}
\end{tikzpicture}
\end{document}


I am also wondering if you want to have the block structure as in your question, if so, please let me know. (BTW, I think it would be better if you move your answer to your question.) Just in case:

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{plotmarks,calc}
\begin{document}
\pgfsetplotmarksize{0.05cm}
\begin{tikzpicture}
%draws the outer box
\draw (0.15,-0.15) coordinate (br) rectangle (-3.95,3.9) coordinate (tl);

\foreach \x in  {0,...,25}
{\foreach \y in  {0,...,25}
{\pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)<2,0,1)}
\ifnum\Z=0
\node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
\else
\pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)==4,0,1)}
\ifnum\Z=0
\node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
\fi
\fi}
}
\fill[white] ([xshift=1mm,yshift=1mm]br -| tl) rectangle ([xshift=0.2mm]$(br)!0.5!(tl)$);
\fill[white] ([xshift=-1mm,yshift=-1mm]br |- tl) rectangle ([xshift=0.2mm]$(br)!0.5!(tl)$);
\end{tikzpicture}
\end{document}


• thanks for the answer with additional answer for the block structure which I'll eventually need ! Commented May 20, 2018 at 20:13
• @gajendra You're welcome. It will be cleaner not to draw these marks (by adjusting the ifthenelse thingies) in the general case, and also more elegant...
– user121799
Commented May 20, 2018 at 20:26

Five years ago I wrote this solution. Sorry for bad results but I deleted input rows to stay within 30000 rows

% !TEX program = LuaLaTeX

%\documentclass[a4paper]{report}
%\usepackage[margin=2cm]{geometry}

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{calc}

\def\LuaCalc#1{\directlua{tex.print(#1)}}

\def\pixels#1#2#3#4[#5]{
% #1 = pixel list
% #2 = X resolution
% #3 = Y resolution
% #4 = south west vertex
% #5 = grid key-values
\path[step=1,#5] (#4)grid(#2,#3);

\foreach \pixel in {#1}
{
\edef\column{\LuaCalc{math.ceil(\pixel/#3)}}

\ifnum\column>1
\edef\row{\LuaCalc{#3-\pixel+(\column-1)*#3}}
\else
\edef\row{\LuaCalc{#3-\pixel}}
\fi

\ifnum#2<\column
\edef\maxPix{\LuaCalc{#2*#3}}
\errmessage{Pixel value is \pixel. The maximum must be \maxPix}
\fi

\fill (\column-1,\row)rectangle++(1,1);
}
}

\begin{document}
\begin{tikzpicture}[x=.2mm,y=.2mm]
\pixels{72,78,93,94,98,99,%
120,121,122,136,137,138,%
139,141,143,185,272,273,%
277,278,279,294,295,298,%
309,320,322,337,338,341,%
342,343,385,473,474,475,%
476,477,494,495,496,497,%
498,499,509,510,511,512,%
518,519,520,522,536,537,%
538,539,542,543,585,586,%
587,657,658,672,673,677,%
689,694,695,696,709,711,%
719,720,722,736,742,747,%
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895,908,909,911,912,941,%
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3940,3941,3952,3954,3958,3964,%
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3996,3998,3999,4000,4001,4033,%
4034,4050,4051,4054,4055,4070,%
4071,4097,4113,4118,4119,4123,%
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4201,4241,4250,4251,4253,4254,%
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5820,5821,5822,5823,5824,5838,%
5839,5840,5856,5857,5870,5872,%
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5901,5903,5906,5932,5933,5935,%
5936,5947,5982,5986,5988,5999,%
6015,6016,6017,6018,6023,6024,%
6038,6039,6040,6054,6055,6056,%
6057,6069,6070,6071,6072,6075,%
6076,6078,6079,6101,6102,6103,%
6104,6105,6106,6129,6133,6134,%
6135,6136,6146,6147,6182,6183,%
6186,6188,6216,6240,6241,6255,%
6271,6272,6275,6276,6295,6329,%
6330,6333,6334,6335,6386,6388,%
6415,6416,6417,6471,6472,6476,%
6477,6492,6493,6494,6495,6529,%
6530,6534,6585,6586,6587,6588,%
6589,6613,6614,6615,6617,6667,%
6668,6669,6670,6671,6672,6685,%
6686,6692,6695,6729,6730,6782,%
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6817,6867,6868,6869,6871,6886,%
6892,6893,6894,6895,6928,6929,%
6939,6980,6981,6982,6983,6985,%
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7062,7066,7067,7068,7069,7078,%
7079,7086,7095,7096,7128,7139,%
7140,7141,7179,7180,7185,7196,%
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7215,7261,7279,7286,7289,7291,%
7294,7295,7296,7328,7330,7331,%
7337,7340,7380,7383,7384,7385,%
7386,7387,7396,7415,7416,7433,%
7434,7435,7460,7461,7479,7486,%
7487,7488,7489,7490,7491,7495,%
7496,7525,7526,7527,7528,7529,%
7530,7531,7537,7540,7541,7580,%
7581,7585,7587,7596,7597,7615,%
7616,7617,7618,7631,7632,7633,%
7634,7646,7647,7648,7649,7660,%
7661,7666,7668,7674,7675,7676,%
7677,7678,7679,7680,7681,7686,%
7691,7692,7694,7695,7720,7725,%
7730,7731,7737,7740,7781,7784,%
7785,7787,7797,7815,7816,7818,%
7831,7846,7848,7849,7850,7851,%
7861,7862,7864,7865,7866,7867,%
7868,7876,7877,7878,7879,7881,%
7882,7891,7892,7918,7919,7920,%
7921,7925,7926,7930,7937,7938,%
7939,7940,7984,7985,7987,7997,%
8015,8029,8031,8050,8061,8064,%
8065,8066,8067,8075,8076,8077,%
8081,8091,8092,8093,8118,8121,%
8125,8126,8128,8129,8130,8131,%
8132,8138,8139,8173,8174,8181,%
8182,8183,8184,8197,8198,8215,%
8216,8225,8226,8227,8228,8229,%
8230,8231,8232,8233,8236,8261,%
8267,8281,8288,8291,8318,8326,%
8327,8328,8329,8331,8337,8338,%
8339,8372,8373,8374,8381,8383,%
8397,8405,8406,8425,8427,8428,%
8430,8431,8432,8433,8434,8435,%
8436,8437,8438,8439,8460,8461,%
8462,8465,8466,8467,8476,8477,%
8478,8479,8480,8481,8488,8489,%
8516,8517,8518,8527,8537,8538,%
8539,8547,8548,8549,8572,8573,%
8581,8583,8601,8602,8603,8604,%
8605,8632,8633,8635,8660,8661,%
8662,8667,8688,8716,8717,8727,%
8733,8734,8738,8739,8740,8747,%
8772,8773,8774,8801,8805,8808,%
8809,8810,8832,8835,8860,8862,%
8867,8888,8889,8890,8891,8892,%
8893,8894,8917,8925,8926,8927,%
8928,8934,8935,8936,8937,8938,%
8939,8940,8947,8948,8949,8950,%
8971,8972,8974,8975,8979,8980,%
8981,8983,8984,9004,9005,9006,%
9009,9010,9031,9032,9035,9036,%
9037,9038,9062,9067,9089,9092,%
9094,9125,9127,9128,9134,9136,%
9139,9140,9141,9147,9148,9149,%
9150,9155,9180,9184,9185,9204,%
9205,9206,9208,9209,9210,9219,%
9232,9233,9234,9235,9236,9253,%
9254,9255,9262,9267,9268,9294,%
9295,9319,9325,9326,9327,9328,%
9329,9334,9336,9337,9338,9339,%
9340,9347,9355,9366,9368,9376,%
9377,9379,9380,9382,9383,9384,%
9385,9410,9417,9418,9419,9432,%
9452,9453,9455,9456,9457,9458,%
9462,9467,9494,9519,9520,9522,%
9527,9528,9529,9537,9539,9547,%
9555,9566,9567,9568,9576,9577,%
9578,9579,9582,9585,9609,9610,%
9611,9617,9618,9630,9631,9632,%
9635,9636,9637,9638,9644,9645,%
9648,9657,9662,9667,9668,9669,%
9693,9694,9703,9705,9719,9722,%
9723,9725,9726,9727,9729,9737,%
9747,9748,9753,9754,9755,9766,%
9768,9782,9783,9785,9807,9809,%
9817,9818,9819,9831,9832,9838,%
9843,9844,9845,9848,9857,9859,%
9860,9861,9862,9871,9893,9903,%
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9919,9920,9921,9922,9923,9924,%
9925,9926,9927,9948,9951,9952,%
9953,9968,9982,9983,9984,9985,%
9995,9996,10005,10006,10007,10019,%
10031,10032,10038,10039,10045,10046,%
10047,10048,10051,10059,10061,10071,%
10072,10080,10081,10082,10084,10087,%
10105,10106,10107,10112,10118,10119,%
10120,10124,10127,10129,10147,10148,%
10149,10152,10153,10154,10156,10167,%
10168,10182,10192,10193,10194,10195,%
10196,10197,10201,10202,10203,10204,%
10205,10207,10217,10218,10219,10225,%
10239,10245,10248,10251,10254,10261,%
10262,10271,10272,10280,10281,10282,%
10283,10284,10287,10292,10293,10305,%
10311,10312,10313,10314,10315,10318,%
10319,10327,10328,10329,10330,10331,%
10332,10352,10354,10355,10356,10357,%
10367,10368,10388,10393,10394,10396,%
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10440,10448,10449,10450,10451,10452,%
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10928,10950,10951,10952,10956,10966,%
10967,10968,10969,10986,10987,10991,%
10992,10996,10997,11006,11007,11009,%
11024,11027,11032,11034,11046,11074,%
11075,11077,11088,11089,11090,11091,%
11097,11098,11101,11102,11103,11104,%
11105,11127,11128,11154,11156,11160,%
11186,11187,11191,11192,11193,11196,%
11204,11205,11206,11207,11208,11209,%
11223,11224,11231,11232,11234,11246,%
11247,11248,11268,11274,11277,11290,%
11291,11297,11298,11302,11304,11305,%
11327,11332,11354,11355,11356,11357,%
11359,11360,11362,11363,11364,11369,%
11386,11387,11388,11391,11393,11394,%
11395,11396,11409,11429,11430,11431,%
11432,11444,11445,11446,11447,11448,%
11453,11465,11466,11467,11468,11469,%
11474,11475,11477,11496,11497,11498,%
11527,11532,11533,11534,11554,11555,%
11556,11557,11558,11559,11560,11561,%
11562,11563,11564,11569,11570,11572,%
11587,11588,11593,11594,11596,11608,%
11609,11610,11629,11631,11644,11648,%
11653,11658,11659,11663,11664,11665,%
11669,11674,11675,11676,11677,11690,%
11695,11696,11697,11698,11699,11723,%
11724,11725,11734,11754,11755,11756,%
11757,11761,11764,11765,11769,11772,%
11773,11774,11775,11776,11777,11778,%
11779,11780,11787,11788,11796,11809,%
11829,11830,11831,11843,11844,11848,%
11852,11853,11858,11859,11863,11864,%
11865,11867,11868,11869,11874,11888,%
11889,11890,11891,11897,11925,11926,%
11927,11928,11931,11932,11933,11934,%
11935,11960,11961,11963,11964,11965,%
11966,11967,11968,11969,11977,11980,%
11988,11989,12028,12029,12033,12034,%
12035,12036,12037,12038,12039,12042,%
12043,12044,12045,12046,12048,12053,%
12058,12059,12067,12069,12070,12089,%
12091,12124,12125,12127,12131,12132,%
12158,12159,12160,12164,12165,12166,%
12167,12168,12169,12177,12188,12228,%
12229,12239,12242,12244,12245,12251,%
12252,12253,12257,12258,12259,12270,%
118890,118891,118892,118893,118906,118907,%
118908,118912,118930,118940,118941,118942,%
118944,118945,118947,118949,118972,118973,%
118986,118995,118996,119001,119002,119003,%
119013,119014,119018,119035,119036,119038,%
119064,119065,119066,119067,119070,119089,%
119090,119091,119093,119106,119107,119108,%
119109,119110,119111,119112,119130,119131,%
119136,119141,119144,119145,119149,119150,%
119172,119186,119187,119196,119198,119199,%
119202,119203,119204,119214,119233,119234,%
119235,119236,119237,119238,119240,119241,%
119242,119243,119244,119265,119270,119289,%
119310,119330,119331,119336,119337,119338,%
119341,119343,119344,119350,119387,119388,%
119389,119398,119399,119401,119402,119403,%
119404,119405,119414,119436,119437,119443,%
119444,119445,119467,119468,119469,119470,%
119471,119489,119493,119494,119495,119496,%
119497,119531,119536,119537,119538,119543,%
119555,119556,119589,119594,119595,119596,%
119597,119598,119599,119600,119604,119605,%
119614,119615,119616,119617,119637,119642,%
119643,119644,119645,119671,119672,119673,%
119674,119693,119696,119731,119736,119737,%
119740,119742,119743,119751,119752,119753,%
119754,119755,119756,119757,119787,119788,%
119789,119794,119795,119797,119803,119804,%
119814,119815,119836,119837,119839,119840,%
119841,119842,119844,119845,119871,119873,%
119893,119894,119896,119897,119931,119937,%
119940,119941,119942,119943,119955,119956,%
119957,119987,119989,119990,119991,119992,%
119994,119995,119997,119998,119999,120000%
}{600}{200}{0,0}[]

\end{tikzpicture}
\end{document}