| Summary: | A probabilistic approach is developed for the exact solution <i>u</i> to a deterministic partial differential equation as well as for its associated approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> performed by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula> Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution <i>u</i> relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula>. We, thus, fill this knowledge gap by considering the exact solution <i>u</i> together with its corresponding approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> as random variables. By a method of consequence, any function where <i>u</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mrow><mi>h</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></semantics></math></inline-formula> Sobolev spaces and the corresponding a priori estimates of the exact solution <i>u</i> and its approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> in order to consider their respective <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></mrow></semantics></math></inline-formula>-norm as a random variable, as well as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></semantics></math></inline-formula> approximation error with regards to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula> finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub><mo>,</mo><mrow><mo stretchy="false">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>.
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