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So it is natural to replace the usual$$L^2(Q)$$<\/jats:tex-math>L<\/mml:mi>2<\/mml:mn><\/mml:msup>(<\/mml:mo>Q<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>norm regularization by the energy regularization in the$$L^2(0,T;H^{-1}(\\Omega ))$$<\/jats:tex-math>L<\/mml:mi>2<\/mml:mn><\/mml:msup>(<\/mml:mo>0<\/mml:mn>,<\/mml:mo>T<\/mml:mi>\u037e<\/mml:mo>H<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup>(<\/mml:mo>\u03a9<\/mml:mi>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>norm. We derive new a priori estimates for the error$$\\Vert \\widetilde{u}_{\\varrho h} - \\overline{u}\\Vert _{L^2(Q)}$$<\/jats:tex-math>\u2016<\/mml:mo><\/mml:mrow>u<\/mml:mi>~<\/mml:mo><\/mml:mover>\u03f1<\/mml:mi>h<\/mml:mi><\/mml:mrow><\/mml:msub>-<\/mml:mo>u<\/mml:mi>\u00af<\/mml:mo><\/mml:mover>\u2016<\/mml:mo><\/mml:mrow>L<\/mml:mi>2<\/mml:mn><\/mml:msup>(<\/mml:mo>Q<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>between the computed state$$\\widetilde{u}_{\\varrho h}$$<\/jats:tex-math>u<\/mml:mi>~<\/mml:mo><\/mml:mover>\u03f1<\/mml:mi>h<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and the desired state$$\\overline{u}$$<\/jats:tex-math>u<\/mml:mi>\u00af<\/mml:mo><\/mml:mover><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in terms of the regularization parameter$$\\varrho $$<\/jats:tex-math>\u03f1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and the space-time finite element mesh sizeh<\/jats:italic>, and depending on the regularity of the desired state$$\\overline{u}$$<\/jats:tex-math>u<\/mml:mi>\u00af<\/mml:mo><\/mml:mover><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. These new estimates lead to the optimal choice$$\\varrho = h^2$$<\/jats:tex-math>\u03f1<\/mml:mi>=<\/mml:mo>h<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The approximate state$$\\widetilde{u}_{\\varrho h}$$<\/jats:tex-math>u<\/mml:mi>~<\/mml:mo><\/mml:mover>\u03f1<\/mml:mi>h<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes forQ<\/jats:italic>. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions. 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