Let D be the rectangle
and let C be the boundary of D.
The operator
is the usual Laplacian. The problem, determine a function u(x, y)
such that
is called a Poisson problem.
To approximate u(x, y) numerically, use the grid
.
with
![{\displaystyle x_{i}=i\,h,\,y_{i}=j\ k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef6822fc4e5a868c9387f9bfe294d3f66294647)
and
![{\displaystyle h=a\,/\,(m+1),k=b\,/\,(n+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f0fa0225b49c706b87bc7e50c080d419efb071)
The second partial derivative
![{\displaystyle {\partial ^{2} \over \partial x^{2}}\ u(x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b17f1d3a3d7bdcc50a2839a95a7b7348f04a712)
can be approximated on the grid by difference quotients
.
These difference quotients are given by
.
.
.
The second partial derivative
![{\displaystyle {\partial ^{2} \over \partial y^{2}}\ u(x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dea4847236935b4aeeb58786af9efeb696ff6030)
can be approximated on the grid by difference quotients
.
These difference quotients are given by
.
.
.
The difference quotients
are third order accurate with truncation errors:
with
,
for some
,
,
for some ![{\displaystyle -2<\phi {_{1}^{(i,\,j)}}<1\,,\quad -1<\phi {_{2}^{(i,\,j)}}<2\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24785134b17a3aa74c20cd39852eef8d2f121393)
When
is
continuous, these estimates also hold.
with
![{\displaystyle M_{x}^{6}(x_{i},\,y_{j})\;=\;{\frac {8}{3}}{\partial ^{6} \over \partial x^{6}}u(x_{i}+\phi {_{1}^{(i,\,j)}}\,h,\;y_{j})-{\frac {32}{3}}{\partial ^{6} \over \partial x^{6}}u(x_{i}+\phi {_{2}^{(i,\,j)}}\,h,\;y_{j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8552399173e3c70ee457a8c03389f620e8f06a77)
for some
![{\displaystyle {\text{for}}\ \ i\ =\ 2,\ 3,\ \ldots \,,\ m-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3d6b208e5ffb50c74eece73cb1f2a8f69ccf97)
The case for
is
,
for some
.
The difference quotients
are third order accurate with truncation errors:
with
,
for some
,
,
for some ![{\displaystyle -2<\psi {_{1}^{(i,\,j)}}<1\,,\quad -1<\psi {_{2}^{(i,\,j)}}<2\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa6210132212f0dbe0bc872b69cb338a079608a)
When
is
continuous, these estimates also hold.
with
![{\displaystyle M_{y}^{6}(x_{i},\,y_{j})\;=\;{\frac {8}{3}}{\partial ^{6} \over \partial y^{6}}u(x_{i}+\psi {_{1}^{(i,\,j)}}\,k,\;y_{j})-{\frac {32}{3}}{\partial ^{6} \over \partial y^{6}}u(x_{i}+\psi {_{2}^{(i,\,j)}}\,k,\;y_{j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03acc053f274ee6917f03936cabff142bb867e2d)
for some
![{\displaystyle {\text{for}}\ \ j\ =\ 2,\ 3,\ \ldots \,,\ n-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db612ede158e65559d48d2dbaab7b37104fe3c59)
The case for
is
,
for some
.
The Laplacian
then can be
approximated on the interior of the grid by
The truncation error
is given by
.
For the grid vector
define the finite difference operations
by the following.
.
.
.
.
.
.
To simulate the problem (1.0) let
Then solve the non-singular linear system
;
for the remaining
The error
,
satisfies
.
for
,
and
.
The truncation error estimates for
are done under the assumption that
is
sufficiently smooth so that
is continuous.
For notational convenience let
![{\displaystyle g(x)\;=\;u(x,\;y_{j}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b31be1bd057694182b176043f06253113c4737c9)
Expand
in it's Taylor expansion about
,
.
where
is some number between
and
. Then
![{\displaystyle {\begin{aligned}&{\partial _{h}^{2} \over \partial _{h}x^{2}}\,u(x_{1},\,y_{j})\;=\\&{{\big (}-{\frac {1}{12}}g(x_{1}+3\,h)+{\frac {1}{3}}g(x_{1}+2\,h)+{\frac {1}{2}}g(x_{1}+h)-{\frac {5}{3}}g(x_{1})+{\frac {11}{12}}g(x_{1}-h){\big )} \over \ h^{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/617a81d2ac5f2617729a37eddace70fff5bd01a5)
![{\displaystyle {\begin{aligned}&=\;g(x_{1}){(-{\frac {1}{12}}+{\frac {1}{3}}+{\frac {1}{2}}-{\frac {5}{3}}+{\frac {11}{12}}) \over \ h^{2}}\\&+\;g^{(1)}(x_{1}){(-{\frac {1}{12}}(3\,h)+{\frac {1}{3}}(2\,h)+{\frac {1}{2}}(h)-{\frac {5}{3}}(0)+{\frac {11}{12}}(-h)) \over \ h^{2}}\\&+\;{\frac {1}{2}}g^{(2)}(x_{1}){(-{\frac {1}{12}}(3\,h)^{2}+{\frac {1}{3}}(2\,h)^{2}+{\frac {1}{2}}(h)^{2}-{\frac {5}{3}}(0)^{2}+{\frac {11}{12}}(-h)^{2}) \over \ h^{2}}\\&+\;{\frac {1}{6}}g^{(3)}(x_{1}){(-{\frac {1}{12}}(3\,h)^{3}+{\frac {1}{3}}(2\,h)^{3}+{\frac {1}{2}}(h)^{3}-{\frac {5}{3}}(0)^{3}+{\frac {11}{12}}(-h)^{3}) \over \ h^{2}}\\&+\;{\frac {1}{24}}g^{(4)}(x_{1}){(-{\frac {1}{12}}(3\,h)^{4}+{\frac {1}{3}}(2\,h)^{4}+{\frac {1}{2}}(h)^{4}-{\frac {5}{3}}(0)^{4}+{\frac {11}{12}}(-h)^{4}) \over \ h^{2}}\\&+\;{\frac {1}{120}}{(-{\frac {1}{12}}g^{(5)}(z_{1})(3\,h)^{5}+{\frac {1}{3}}g^{(5)}(z_{2})(2\,h)^{5}+{\frac {1}{2}}g^{(5)}(z_{3})(h)^{5}-{\frac {5}{3}}(0)^{5} \over \ h^{2}}\\&\quad \quad \quad \quad {+{\frac {11}{12}}g^{(5)}(z_{4})(-h)^{5}) \over \ h^{2}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c735981fc660005e8ab56e98cd06b6920ca65e)
where
.
Since
![{\displaystyle {\frac {67}{6}}{\underset {0\;\leq \;\phi \;\leq \;2}{\min(g^{(5)}(x_{1}\;+\;\phi \,h))}}\;\leq \;{\frac {32}{3}}g^{(5)}(z_{2})+{\frac {1}{2}}g^{(5)}(z_{3})\;\leq \;{\frac {67}{6}}{\underset {0\;\leq \;\phi \;\leq \;2}{\max(g^{(5)}(x_{1}\;+\;\phi \,h))}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a716c5ff15e42f6e813a1742f1f5e5daf38a2705)
![{\displaystyle {\frac {254}{12}}{\underset {-1\;\leq \;\phi \;\leq \;3}{\min(g^{(5)}(x_{1}\;+\;\phi \,h))}}\;\leq \;{\frac {81}{4}}g^{(5)}(z_{1})+{\frac {11}{12}}g^{(5)}(z_{4})\;\leq \;{\frac {254}{12}}{\underset {-1\;\leq \;\phi \;\leq \;3}{\max(g^{(5)}(x_{1}\;+\;\phi \,h))}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f5f5f16566e77909d5f770ffd86336a5e10608)
from the intermediate value property
![{\displaystyle {\frac {32}{3}}g^{(5)}(z_{2})+{\frac {1}{2}}g^{(5)}(z_{3})\;=\;{\frac {67}{6}}\,g^{(5)}(x_{1}\;+\;\phi _{1}\,h)),\quad {\text{for}}\;0\;<\;\phi _{1}\,,\;<\;2\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c36e6aa277997ea5f9d49ffd5184d964f4397b)
.
This gives
![{\displaystyle {\begin{aligned}&{\partial _{h}^{2} \over \partial _{h}x^{2}}\,u(x_{1},\,y_{j})\;=\;g^{(2)}(x_{1})+{h^{3} \over \ 120}{\big (}{\frac {67}{6}}\,g^{(5)}(x_{1}+\phi _{1}\,h)-{\frac {254}{12}}\,g^{(5)}(x_{1}+\phi _{2}\,h){\big )}\\&\;=\;{\partial ^{2} \over \partial x^{2}}\,u(x_{1},\,y_{j})+{h^{3} \over \ 120}{\big (}{\frac {67}{6}}{\partial ^{5} \over \partial x^{5}}u(x_{1}+\phi _{1}\,h,\;y_{j})-{\frac {254}{12}}{\partial ^{5} \over \partial x^{5}}u(x_{1}+\phi _{2}\,h,\;y_{j}){\big )}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6636db90b88b704b96cdbfa78203a4379a53ea98)
which is
![{\displaystyle \tau _{x}(x_{1},\,y_{j})\;=\;{h^{3} \over \ 120}M_{x}^{5}(x_{1},\,y_{j}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b9280f76368a298bd8eb0a429366da54abb6ea)
For ![{\displaystyle i\;=\;2,\;3,\;\ldots \,,\;m-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b759980dd35b5c94ec57c0226cc08ee6fadd20)
![{\displaystyle {\begin{aligned}&{\partial _{h}^{2} \over \partial _{h}x^{2}}\,u(x_{i},\,y_{j})\;=\\&{{\big (}-{\frac {1}{12}}g(x_{i}+2\,h)+{\frac {4}{3}}g(x_{i}+h)-{\frac {5}{2}}g(x_{i})+{\frac {4}{3}}g(x_{i}-h)-{\frac {1}{12}}g(x_{i}-2\,h){\big )} \over \ h^{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/923b12dd1936e72da058459e37b66346eb3399d7)
![{\displaystyle {\begin{aligned}&=\;g(x_{i}){(-{\frac {1}{12}}+{\frac {4}{3}}-{\frac {5}{2}}+{\frac {4}{3}}-{\frac {1}{12}}) \over \ h^{2}}\\&+\;g^{(1)}(x_{i}){(-{\frac {1}{12}}(2\,h)+{\frac {4}{3}}(h)-{\frac {5}{2}}(0)+{\frac {4}{3}}(-h)-{\frac {1}{12}}(-2\,h)) \over \ h^{2}}\\&+\;{\frac {1}{2}}g^{(2)}(x_{i}){(-{\frac {1}{12}}(2\,h)^{2}+{\frac {4}{3}}(h)^{2}-{\frac {5}{2}}(0)^{2}+{\frac {4}{3}}(-h)^{2}-{\frac {1}{12}}(-2\,h)^{2}) \over \ h^{2}}\\&+\;{\frac {1}{6}}g^{(3)}(x_{i}){(-{\frac {1}{12}}(2\,h)^{3}+{\frac {4}{3}}(h)^{3}-{\frac {5}{2}}(0)^{3}+{\frac {4}{3}}(-h)^{3}-{\frac {1}{12}}(-2\,h)^{3}) \over \ h^{2}}\\&+\;{\frac {1}{24}}g^{(4)}(x_{i}){(-{\frac {1}{12}}(2\,h)^{4}+{\frac {4}{3}}(h)^{4}-{\frac {5}{2}}(0)^{4}+{\frac {4}{3}}(-h)^{4}-{\frac {1}{12}}(-2\,h)^{4}) \over \ h^{2}}\\&+\;{\frac {1}{120}}{(-{\frac {1}{12}}g^{(5)}(z_{1})(2\,h)^{5}+{\frac {4}{3}}g^{(5)}(z_{2})(h)^{5}-{\frac {5}{2}}(0)^{5}+{\frac {4}{3}}g^{(5)}(z_{3})(-h)^{5} \over \ h^{2}}\\&\quad \quad \quad \quad {-{\frac {1}{12}}g^{(5)}(z_{4})(-2\,h)^{5}) \over \ h^{2}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db23e2c2f42fada240ac530557f475052549a82)
where
.
Reasoning as before, combining terms with like signs and using the intermediate value property,
![{\displaystyle {\frac {4}{3}}g^{(5)}(z_{2})+{\frac {8}{3}}g^{(5)}(z_{4})\;=\;4\,g^{(5)}(x_{i}\;+\;\phi _{1}\,h)),\quad {\text{for}}\;-2\;<\;\phi _{1}\,,\;<\;1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a00fdf0dc13117178798ca7e702ddd62a7981842)
.
This gives
![{\displaystyle {\begin{aligned}&{\partial _{h}^{2} \over \partial _{h}x^{2}}\,u(x_{i},\,y_{j})\;=\;g^{(2)}(x_{i})+{h^{3} \over \ 120}{\big (}{\frac {67}{6}}\,g^{(5)}(x_{i}+\phi _{1}\,h)-{\frac {254}{12}}\,g^{(5)}(x_{i}+\phi _{2}\,h){\big )}\\&\;=\;{\partial ^{2} \over \partial x^{2}}\,u(x_{i},\,y_{j})+{h^{3} \over \ 120}{\big (}4{\partial ^{5} \over \partial x^{5}}u(x_{i}+\phi _{1}\,h,\;y_{j})-4{\partial ^{5} \over \partial x^{5}}u(x_{i}+\phi _{2}\,h,\;y_{j}){\big )}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0423ba05fd05ad36b620f97cf80f2d632faa3ed)
which is
![{\displaystyle \tau _{x}(x_{i},\,y_{j})\;=\;{h^{3} \over \ 120}M_{x}^{5}(x_{i},\,y_{j}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee289a906e7a363ca79d6690542e59f94a12f2d7)
Under the assumption that
is continuous, in the preceding argument, the expression
can be replaced by
This gives
![{\displaystyle {\begin{aligned}&{\partial _{h}^{2} \over \partial _{h}x^{2}}\,u(x_{i},\,y_{j})\;=\;\\&\;=\;{\partial ^{2} \over \partial x^{2}}\,u(x_{i},\,y_{j})+{h^{4} \over \ 720}{\big (}{\frac {8}{3}}{\partial ^{6} \over \partial x^{6}}u(x_{i}+\phi _{1}\,h,\;y_{j})-{\frac {32}{3}}{\partial ^{6} \over \partial x^{6}}u(x_{i}+\phi _{2}\,h,\;y_{j}){\big )}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fea65d9f1d2d7037ee59da4cceb92047e991ebe)
with
which is
![{\displaystyle \tau _{x}(x_{i},\,y_{j})\;=\;{h^{4} \over \ 720}M_{x}^{6}(x_{i},\,y_{j}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97b967d184f6544ec4c2351e94df902cfa420177)
The remaining truncation error estimates are done in the same way.
Let the error
be defined by
.
is the solution of the finite difference scheme
(xx) and
is the solution to (1.0).
Since
we get that
.
Next it will be shown that the operator
is positive definite for
,
in particular that
,
with
.
Begin with
![{\displaystyle \sum _{i=1}^{m}\sum _{j=1}^{n}e_{i,\,j}\,(-\Delta _{i,\,j}\,e)\;=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2750a4942ab81f54ebb1a3beeb50a03c321fd3)
![{\displaystyle \sum _{i=1}^{m}\sum _{j=1}^{n}e_{i,\,j}\,(-({\partial _{h}^{2} \over \partial _{h}x^{2}})_{i,\,j}\,e-({\partial _{k}^{2} \over \partial _{k}y^{2}})_{i,\,j}\,e)\;=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e93e11c3a13b52519f8ad0f9bcb691584eefe906)
![{\displaystyle \sum _{j=1}^{n}\sum _{i=1}^{m}e_{i,\,j}\,(-({\partial _{h}^{2} \over \partial _{h}x^{2}})_{i,\,j}\,e)\;+\;\sum _{i=1}^{m}\sum _{j=1}^{n}e_{i,\,j}\,(-({\partial _{k}^{2} \over \partial _{k}y^{2}})_{i,\,j}\,e)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f199c63c7fdafc67990547e8783a00f6391a0f94)
The sum
will be estimated first.
.
.
.
The summation by parts formula is now stated
so it can be used.
![{\displaystyle \sum _{i=1}^{m}w_{i}\,(v_{i}-v_{i-1})=w_{m}\,v_{m}-w_{0}\,v_{0}-\sum _{i=0}^{m-1}v_{i}\,(w_{i+1}-w_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/713820980c889ffcbf7f6d5b8badd5c9bb3b7681)
![{\displaystyle Now,\quad -h^{2}({\partial _{h}^{2} \over \partial _{h}x^{2}})_{i,\,j}\ e\ =v_{i,\,j}-v_{i-1,\,j},\quad {\text{with}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe36e6c93bfd0ea597130593ae8198293c1f935)
![{\displaystyle v_{0,\,j}\;=\;(-{\frac {1}{12}}\,e_{4,\,j}+{\frac {5}{12}}\,e_{3,\,j}-{\frac {3}{4}}\,e_{2,\,j}-{\frac {5}{12}}\,e_{1,\,j}+{\frac {5}{6}}\,e_{0,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae42b17715f0847090e2679ec5099ae99764a52b)
![{\displaystyle =\;-{\frac {1}{12}}\,(e_{4,\,j}-e_{3,\,j})+{\frac {1}{3}}\,(e_{3,\,j}-e_{2,\,j})-{\frac {5}{12}}\,(e_{2,\,j}-e_{1,\,j})-{\frac {5}{6}}\,(e_{1,\,j}-e_{0,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/254b73ffdec51f3d95f130bf819f7041d79ead91)
![{\displaystyle {\begin{aligned}v_{i,\,j}\;&=\;({\frac {1}{12}}\,e_{i+2,\,j}-{\frac {5}{4}}\,e_{i+1,\,j}+{\frac {5}{4}}\,e_{i,\,j}-{\frac {1}{12}}\,e_{i-1,\,j})\\\;&=\;{\frac {1}{12}}\,(e_{i+2,\,j}-e_{i+1,\,j})-{\frac {7}{6}}\,(e_{i+1,\,j}-e_{i,\,j})+{\frac {1}{12}}\,(e_{i,\,j}-e_{i-1,\,j})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a72b09200af0de23b6701daac62086894373b1)
.
![{\displaystyle v_{m,\,j}\;=\;(-{\frac {5}{6}}\,e_{m+1,\,j}+{\frac {5}{12}}\,e_{m,\,j}+{\frac {3}{4}}\,e_{m-1,\,j}+{\frac {1}{4}}\,e_{m-2,\,j}+{\frac {1}{12}}\,e_{m-3,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97324f79d8a98e464764f811867d86174f214509)
![{\displaystyle =\;-{\frac {5}{6}}\,(e_{m+1,\,j}-e_{m,\,j})-{\frac {5}{12}}\,(e_{m,\,j}-e_{m-1,\,j})+{\frac {1}{3}}\,(e_{m-1,\,j}-e_{m-2,\,j})-{\frac {1}{12}}\,(e_{m-2,\,j}-e_{m-3,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0709952590b4b5be0037f56fdb0bae6bf9fb2709)
![{\displaystyle =e_{m,\,j}\,v_{m,\,j}-e_{0,\,j}\,v_{0,\,j}-\sum _{i=0}^{m-1}v_{i,\,j}\,(e_{i+1,\,j}-e_{i,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be7ff84573541b1193c2c61177d7dee0f74dd9f)
Taking into account that
it follows
![{\displaystyle \sum _{i=1}^{m}e_{i,\,j}\,(-h^{2}({\partial _{h}^{2} \over \partial _{h}x^{2}})_{i,\,j}\,e)=\;-\sum _{i=0}^{m}v_{i,\,j}\,(e_{i+1,\,j}-e_{i,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/246fa472c7620d6ff621fb2befb95b78076ed58c)
![{\displaystyle =\;-v_{0,\,j}\,(e_{1,\,j}-e_{0,\,j})-\sum _{i=1}^{m-1}v_{i,\,j}\,(e_{i+1,\,j}-e_{i,\,j})-v_{m,\,j}\,(e_{m+1,\,j}-e_{m,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/effbe2d00a75b35cba90e546f70843d96d22dd23)
![{\displaystyle {\begin{aligned}=\;&{\frac {1}{12}}\,(e_{4,\,j}-e_{3,\,j})\,(e_{1,\,j}-e_{0,\,j})-{\frac {1}{3}}\,(e_{3,\,j}-e_{2,\,j})\,(e_{1,\,j}-e_{0,\,j})\\&+{\frac {5}{12}}\,(e_{2,\,j}-e_{1,\,j})\,(e_{1,\,j}-e_{0,\,j})+{\frac {5}{6}}\,(e_{1,\,j}-e_{0,\,j})^{2}\\\;&-{\frac {1}{12}}\sum _{i=1}^{m-1}(e_{i+2,\,j}-e_{i+1,\,j})(e_{i+1,\,j}-e_{i,\,j})+{\frac {7}{6}}\sum _{i=1}^{m-1}(e_{i+1,\,j}-e_{i,\,j})^{2}\\\;&-{\frac {1}{12}}\sum _{i=1}^{m-1}(e_{i,\,j}-e_{i-1,\,j})(e_{i+1,\,j}-e_{i,\,j})+{\frac {5}{6}}\,(e_{m+1,\,j}-e_{m,\,j})^{2}\\\;&+{\frac {5}{12}}\,(e_{m,\,j}-e_{m-1,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\;-{\frac {1}{3}}\,(e_{m-1,\,j}-e_{m-2,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\\\;&+{\frac {1}{12}}\,(e_{m-2,\,j}-e_{m-3,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2005d863e5855e8972017dd6a85c5cce7e316e9)
Collect like terms in the expression immediately above as follows.
![{\displaystyle {\frac {5}{6}}\,(e_{1,\,j}-e_{0,\,j})^{2}+{\frac {7}{6}}\sum _{i=1}^{m-1}(e_{i+1,\,j}-e_{i,\,j})^{2}+{\frac {5}{6}}\,(e_{m+1,\,j}-e_{m,\,j})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6245454c6f9361c8535d78f95a134b58a3d9c92f)
![{\displaystyle =\,{\frac {7}{6}}\sum _{i=0}^{m}(e_{i+1,\,j}-e_{i,\,j})^{2}-{\frac {1}{3}}\,(e_{1,\,j}-e_{0,\,j})^{2}-{\frac {1}{3}}\,(e_{m+1,\,j}-e_{m,\,j})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ccdb8d84eb94d18270cd6500f9d4a9c792a6f68)
![{\displaystyle -{\frac {1}{12}}\sum _{i=1}^{m-1}(e_{i+2,\,j}-e_{i+1,\,j})(e_{i+1,\,j}-e_{i,\,j})-{\frac {1}{12}}\sum _{i=1}^{m-1}(e_{i,\,j}-e_{i-1,\,j})(e_{i+1,\,j}-e_{i,\,j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2efedb5151b30b3c9d08373cdd0b09c9bf6be70)
![{\displaystyle {\begin{aligned}=\,&-{\frac {1}{6}}\sum _{i=2}^{m-1}(e_{i+1,\,j}-e_{i,\,j})(e_{i,\,j}-e_{i-1,\,j})-{\frac {1}{12}}(e_{2,\,j}-e_{1,\,j})(e_{1,\,j}-e_{0,\,j})\\\,&-{\frac {1}{12}}(e_{m+1,\,j}-e_{m,\,j})(e_{m,\,j}-e_{m-1,\,j})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/381f318304d6292db9a9afd752a5bc1bcb4669f6)
Now, rewrite the expression after making cancellations.
![{\displaystyle {\begin{aligned}&{\frac {7}{6}}\sum _{i=0}^{m}(e_{i+1,\,j}-e_{i,\,j})^{2}-{\frac {1}{6}}\sum _{i=2}^{m-1}(e_{i+1,\,j}-e_{i,\,j})(e_{i,\,j}-e_{i-1,\,j})-{\frac {1}{3}}\,(e_{1,\,j}-e_{0,\,j})^{2}\\&-{\frac {1}{3}}\,(e_{m+1,\,j}-e_{m,\,j})^{2}+{\frac {1}{12}}\,(e_{4,\,j}-e_{3,\,j})\,(e_{1,\,j}-e_{0,\,j})\\&-{\frac {1}{3}}\,(e_{3,\,j}-e_{2,\,j})\,(e_{1,\,j}-e_{0,\,j})+{\frac {1}{3}}\,(e_{2,\,j}-e_{1,\,j})\,(e_{1,\,j}-e_{0,\,j})\\&+{\frac {1}{3}}\,(e_{m,\,j}-e_{m-1,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\;-{\frac {1}{3}}\,(e_{m-1,\,j}-e_{m-2,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\\\;&+{\frac {1}{12}}\,(e_{m-2,\,j}-e_{m-3,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88e509d3db8681e1b0564db22a50a3797d48c077)
The following simple inequality will be used to bound terms.
![{\displaystyle {\begin{aligned}(a-b)^{2}&=a^{2}-2\,a\,b+b^{2}\,\geq \,0\\2\,a\,b\,&\leq \,a^{2}+b^{2}\\a\,b\,&\leq \,{\frac {1}{2}}\,(a^{2}+b^{2})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb697e8208a0519a05c2cc00e251ea70839e81f)
and also
.
![{\displaystyle {\begin{aligned}&\sum _{i=2}^{m-1}\left\vert (e_{i+1,\,j}-e_{i,\,j})(e_{i,\,j}-e_{i-1,\,j})\right\vert \;=\;\sum _{i=3}^{m-2}\left\vert (e_{i+1,\,j}-e_{i,\,j})(e_{i,\,j}-e_{i-1,\,j})\right\vert \\&+\;\left\vert (e_{3,\,j}-e_{2,\,j})(e_{2,\,j}-e_{1,\,j})\right\vert \;+\;\left\vert (e_{m,\,j}-e_{m-1,\,j})(e_{m-1,\,j}-e_{m-2,\,j})\right\vert \\&\;\leq \;{\frac {1}{2}}(\sum _{i=3}^{m-2}(e_{i+1,\,j}-e_{i,\,j})^{2}\,+\,\sum _{i=3}^{m-2}(e_{i,\,j}-e_{i-1,\,j})^{2})\\&+\;\left\vert (e_{3,\,j}-e_{2,\,j})(e_{2,\,j}-e_{1,\,j})\right\vert \;+\;\left\vert (e_{m,\,j}-e_{m-1,\,j})(e_{m-1,\,j}-e_{m-2,\,j})\right\vert \\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d47b6a062a2c0a258139b0a73a9cbfae310b90)
![{\displaystyle {\begin{aligned}&=\;\sum _{i=3}^{m-3}(e_{i+1,\,j}-e_{i,\,j})^{2}\;+\;{\frac {1}{2}}\,(e_{3,\,j}-e_{2,\,j})^{2}\;+\;{\frac {1}{2}}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}\\&+\;\left\vert (e_{3,\,j}-e_{2,\,j})(e_{2,\,j}-e_{1,\,j})\right\vert \;+\;\left\vert (e_{m,\,j}-e_{m-1,\,j})(e_{m-1,\,j}-e_{m-2,\,j})\right\vert \\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb19852f6b2ab02aff98bbf657bfb7773ded4d47)
![{\displaystyle {\begin{aligned}&=\;\sum _{i=2}^{m-2}(e_{i+1,\,j}-e_{i,\,j})^{2}\;-\;{\frac {1}{2}}\,(e_{3,\,j}-e_{2,\,j})^{2}\;-\;{\frac {1}{2}}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}\\&+\;\left\vert (e_{3,\,j}-e_{2,\,j})(e_{2,\,j}-e_{1,\,j})\right\vert \;+\;\left\vert (e_{m,\,j}-e_{m-1,\,j})(e_{m-1,\,j}-e_{m-2,\,j})\right\vert \\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/978b99fe4f0b4aba5ed108a81a59a5477460fae9)
![{\displaystyle {\begin{aligned}&\leq \;\sum _{i=2}^{m-2}(e_{i+1,\,j}-e_{i,\,j})^{2}\;-\;{\frac {1}{2}}\,(e_{3,\,j}-e_{2,\,j})^{2}\;-\;{\frac {1}{2}}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}\\&+\;{\frac {1}{2}}\,(\alpha _{1}^{2}\,(e_{3,\,j}-e_{2,\,j})^{2}+(e_{2,\,j}-e_{1,\,j})^{2}\,/\,\alpha _{1}^{2})\\&+\;{\frac {1}{2}}\,(\beta _{1}^{2}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}+(e_{m,\,j}-e_{m-1,\,j})^{2}\,/\,\beta _{1}^{2})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f44864520de0548cc8269035ffe06ef5c1ac7c17)
![{\displaystyle {\begin{aligned}&\;\left\vert (e_{4,\,j}-e_{3,\,j})(e_{1,\,j}-e_{0,\,j})\right\vert \;\leq \;{\frac {1}{2}}\,(\alpha _{2}^{2}\,(e_{4,\,j}-e_{3,\,j})^{2}+(e_{1,\,j}-e_{0,\,j})^{2}\,/\,\alpha _{2}^{2})\\&\;\left\vert (e_{3,\,j}-e_{2,\,j})(e_{1,\,j}-e_{0,\,j})\right\vert \;\leq \;{\frac {1}{2}}\,(\alpha _{3}^{2}\,(e_{3,\,j}-e_{2,\,j})^{2}+(e_{1,\,j}-e_{0,\,j})^{2}\,/\,\alpha _{3}^{2})\\&\;\left\vert (e_{2,\,j}-e_{1,\,j})(e_{1,\,j}-e_{0,\,j})\right\vert \;\leq \;{\frac {1}{2}}\,(\alpha _{4}^{2}\,(e_{2,\,j}-e_{1,\,j})^{2}+(e_{1,\,j}-e_{0,\,j})^{2}\,/\,\alpha _{4}^{2})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9ea1c1635bed433c4c59f2126f1633e6539bac)
![{\displaystyle {\begin{aligned}&\;\left\vert (e_{m,\,j}-e_{m-1,\,j})(e_{m+1,\,j}-e_{m,\,j})\right\vert \\&\quad \quad \quad \leq \;{\frac {1}{2}}\,(\beta _{2}^{2}\,(e_{m,\,j}-e_{m-1,\,j})^{2}+(e_{m+1,\,j}-e_{m,\,j})^{2}\,/\,\beta _{2}^{2})\\&\;\left\vert (e_{m-1,\,j}-e_{m-2,\,j})(e_{m+1,\,j}-e_{m,\,j})\right\vert \\&\quad \quad \quad \leq \;{\frac {1}{2}}\,(\beta _{3}^{2}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}+(e_{m+1,\,j}-e_{m,\,j})^{2}\,/\,\beta _{3}^{2})\\&\;\left\vert (e_{m-2,\,j}-e_{m-3,\,j})(e_{m+1,\,j}-e_{m,\,j})\right\vert \\&\quad \quad \quad \leq \;{\frac {1}{2}}\,(\beta _{4}^{2}\,(e_{m-2,\,j}-e_{m-3,\,j})^{2}+(e_{m+1,\,j}-e_{m,\,j})^{2}\,/\,\beta _{4}^{2})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5507b71c541fdd35a0de07daf71a6c8e925df7e4)
Now, substitute all the inequalities into the expression.
![{\displaystyle {\begin{aligned}&{\frac {7}{6}}\sum _{i=0}^{m}(e_{i+1,\,j}-e_{i,\,j})^{2}-{\frac {1}{6}}\sum _{i=2}^{m-1}(e_{i+1,\,j}-e_{i,\,j})(e_{i,\,j}-e_{i-1,\,j})-{\frac {1}{3}}\,(e_{1,\,j}-e_{0,\,j})^{2}\\&-{\frac {1}{3}}\,(e_{m+1,\,j}-e_{m,\,j})^{2}+{\frac {1}{12}}\,(e_{4,\,j}-e_{3,\,j})\,(e_{1,\,j}-e_{0,\,j})\\&-{\frac {1}{3}}\,(e_{3,\,j}-e_{2,\,j})\,(e_{1,\,j}-e_{0,\,j})+{\frac {1}{3}}\,(e_{2,\,j}-e_{1,\,j})\,(e_{1,\,j}-e_{0,\,j})\\&+{\frac {1}{3}}\,(e_{m,\,j}-e_{m-1,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\;-{\frac {1}{3}}\,(e_{m-1,\,j}-e_{m-2,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\\\;&+{\frac {1}{12}}\,(e_{m-2,\,j}-e_{m-3,\,j})\,(e_{m+1,\,j}-e_{m,\,j})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88e509d3db8681e1b0564db22a50a3797d48c077)
![{\displaystyle {\begin{aligned}&{\frac {7}{6}}\sum _{i=0}^{m}(e_{i+1,\,j}-e_{i,\,j})^{2}-{\frac {1}{6}}\,{\big (}\;\sum _{i=2}^{m-2}(e_{i+1,\,j}-e_{i,\,j})^{2}\;-\;{\frac {1}{2}}\,(e_{3,\,j}-e_{2,\,j})^{2}\;\\&-\;{\frac {1}{2}}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}+\;{\frac {1}{2}}\,(\alpha _{1}^{2}\,(e_{3,\,j}-e_{2,\,j})^{2}+(e_{2,\,j}-e_{1,\,j})^{2}\,/\,\alpha _{1}^{2})\\&+\;{\frac {1}{2}}\,(\beta _{1}^{2}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}+(e_{m,\,j}-e_{m-1,\,j})^{2}\,/\,\beta _{1}^{2})\;{\big )}\\&-{\frac {1}{3}}\,(e_{1,\,j}-e_{0,\,j})^{2}-{\frac {1}{3}}\,(e_{m+1,\,j}-e_{m,\,j})^{2}\\&-{\frac {1}{12}}\,{\big (}{\frac {1}{2}}\,(\alpha _{2}^{2}\,(e_{4,\,j}-e_{3,\,j})^{2}+(e_{1,\,j}-e_{0,\,j})^{2}\,/\,\alpha _{2}^{2}){\big )}\\&-{\frac {1}{3}}\,{\big (}{\frac {1}{2}}\,(\alpha _{3}^{2}\,(e_{3,\,j}-e_{2,\,j})^{2}+(e_{1,\,j}-e_{0,\,j})^{2}\,/\,\alpha _{3}^{2}){\big )}\\&-{\frac {1}{3}}\,{\big (}{\frac {1}{2}}\,(\alpha _{4}^{2}\,(e_{2,\,j}-e_{1,\,j})^{2}+(e_{1,\,j}-e_{0,\,j})^{2}\,/\,\alpha _{4}^{2}){\big )}\\&-{\frac {1}{3}}\,{\big (}{\frac {1}{2}}\,(\beta _{2}^{2}\,(e_{m,\,j}-e_{m-1,\,j})^{2}+(e_{m+1,\,j}-e_{m,\,j})^{2}\,/\,\beta _{2}^{2}){\big )}\\&-{\frac {1}{3}}\,{\big (}{\frac {1}{2}}\,(\beta _{3}^{2}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}+(e_{m+1,\,j}-e_{m,\,j})^{2}\,/\,\beta _{3}^{2}){\big )}\\&-{\frac {1}{12}}\,{\big (}{\frac {1}{2}}\,(\beta _{4}^{2}\,(e_{m-2,\,j}-e_{m-3,\,j})^{2}+(e_{m+1,\,j}-e_{m,\,j})^{2}\,/\,\beta _{4}^{2}){\big )}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a742ed1cf22f78ed3766976d749f9b47790373f)
![{\displaystyle {\begin{aligned}&=\;{\frac {7}{6}}\sum _{i=0}^{m}(e_{i+1,\,j}-e_{i,\,j})^{2}-{\frac {1}{6}}\;\sum _{i=0}^{m}(e_{i+1,\,j}-e_{i,\,j})^{2}\\&\;-\;{\big (}\,{\frac {1}{6}}\,+\,({\frac {1}{24}})\,/\,\alpha _{2}^{2}\,+\,({\frac {1}{6}})\,/\,\alpha _{3}^{2}\,+\,({\frac {1}{6}})\,/\,\alpha _{4}^{2}\,{\big )}\,(e_{1,\,j}-e_{0,\,j})^{2}\\&\;-\;{\big (}\,-{\frac {1}{6}}\,+\,({\frac {1}{12}})\,/\,\alpha _{1}^{2}\,+\,{\frac {1}{6}}\,\alpha _{4}^{2}\,{\big )}\,(e_{2,\,j}-e_{1,\,j})^{2}\\&\;-\;{\big (}\,-{\frac {1}{12}}\,+\,{\frac {1}{12}}\,\alpha _{1}^{2}\,+\,{\frac {1}{6}}\,\alpha _{3}^{2}\,{\big )}\,(e_{3,\,j}-e_{2,\,j})^{2}\\&\;-\;{\big (}\,{\frac {1}{24}}\,\alpha _{2}^{2}\,{\big )}\,(e_{4,\,j}-e_{3,\,j})^{2}\\&\;-\;{\big (}\,{\frac {1}{6}}\,+\,({\frac {1}{6}})\,/\,\beta _{2}^{2}\,+\,({\frac {1}{6}})\,/\,\beta _{3}^{2}\,+\,({\frac {1}{24}})\,/\,\beta _{4}^{2}\,{\big )}\,(e_{m+1,\,j}-e_{m,\,j})^{2}\\&\;-\;{\big (}\,-{\frac {1}{6}}\,+\,({\frac {1}{12}})\,/\,\beta _{1}^{2}\,+\,{\frac {1}{6}}\,\beta _{2}^{2}\,{\big )}\,(e_{m,\,j}-e_{m-1,\,j})^{2}\\&\;-\;{\big (}-{\frac {1}{12}}\,+\,{\frac {1}{12}}\,\beta _{1}^{2}\,+\,{\frac {1}{6}}\,\beta _{3}^{2}\,{\big )}\,(e_{m-1,\,j}-e_{m-2,\,j})^{2}\\&\;-\;{\big (}\,{\frac {1}{24}}\,\beta _{4}^{2}\,{\big )}\,(e_{m-2,\,j}-e_{m-3,\,j})^{2}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/488103dde473647db47ed12c9a2565b3def2c40d)
The choice
![{\displaystyle {\begin{aligned}&\alpha _{1}^{2}=\beta _{1}^{2}=({\sqrt {5}}-1)\,/\,2\,,\alpha _{2}^{2}=\beta _{4}^{2}=8\,,\\&\alpha _{3}^{2}=\alpha _{4}^{2}=\beta _{2}^{2}=\beta _{3}^{2}=(11-{\sqrt {5}})\,/\,4\,,\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3046cbb76fc38727afbcb3c4f4fe6f6a884ed8d0)
bounds all of the coefficients in the
and
by
and yields the long sought inequality
.
and
.
Reasoning in the exact same manner for the dimension in
.
and
.
Applying
![{\displaystyle \sum _{i=1}^{m}\sum _{j=1}^{n}e_{i,\,j}\,(-\Delta _{i,\,j}\,e)\;=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2750a4942ab81f54ebb1a3beeb50a03c321fd3)
![{\displaystyle \sum _{j=1}^{n}\sum _{i=1}^{m}e_{i,\,j}\,(-({\partial _{h}^{2} \over \partial _{h}x^{2}})_{i,\,j}\,e)\;+\;\sum _{i=1}^{m}\sum _{j=1}^{n}e_{i,\,j}\,(-({\partial _{k}^{2} \over \partial _{k}y^{2}})_{i,\,j}\,e)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f199c63c7fdafc67990547e8783a00f6391a0f94)
leads to the inequality
![{\displaystyle \sum _{i=1}^{m}\sum _{j=1}^{n}e_{i,\,j}\,(-\Delta _{i,\,j}\,e)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe98fb8a0dedcb7d27b05a1fa146329bba617e3)
.