Ceramic materials used in solid-oxide fuel cells (SOFCs) are subjected to high thermal stresses which is a result of the unequal thermal expansion coefficient of different layers. As a result, SOFCs are susceptible to failure at elevated temperatures, and therefore the maximum operating temperature is limited. Consequently, the power density is limited as well. Fuel cells with electrodes that have graded compositions in the thickness direction have been investigated to control the thermal expansion. In this study, two-dimensional grading is proposed for the electrodes and compared with the one-dimensional grading in terms of thermal stress and performance. A comprehensive model is developed for high-temperature SOFCs that include the momentum, species, energy, and charge transport equations. Furthermore, the bilinear elastoplastic model is used for the calculation of thermal stresses and failure of solids. Two-dimensional functionally graded electrodes are studied in which the grading is implemented in the thickness and length directions. Results indicate that continuous one-dimensional grading functions reduced thermal stresses by 40% for m = 0.8 compared with conventional electrodes. It also improved the electrochemical performance, as the maximum power density increased by 15%. For the 2D piecewise linear grading function, a further improvement of reducing thermal stresses by an extra 16.5% is obtained. Two-dimensional graded SOFCs can therefore operate at higher temperatures safely in terms of thermal stresses. This creates an opportunity to fabricate high-temperature, compact SOFCs for high-power applications.