| Summary: | In this paper, we propose a new family of probability laws based on the Generalized Beta Prime 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>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Usually, the relative finite element accuracy is based on the comparison of the asymptotic speed of convergence, when the mesh size <i>h</i> goes to zero. The new probability laws we propose here highlight that there exists, depending on <i>h</i>, cases where the <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> finite element is more likely accurate than the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula> element. To confirm this assertion, we highlight, using numerical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities, as determined by the probability law. This illustrates that, when <i>h</i> goes away from zero, a finite element <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> may produce more precise results than a finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>, since the probability of the event “<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><i>is more accurate than</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>” becomes greater than 0.5. In these cases, finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula> is more likely overqualified.
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